Geometric structures and special spinor fields
Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.)
am Fachbereich der Mathematik und Informatik Philipps-Universit¨atMarburg (Hochschulkennziffer 1180)
von Dipl. Math. Jos H¨oll
geboren am 13.05.1984 in Herrenberg
Erstgutachter: Prof. Dr. habil Ilka Agricola (Universit¨atMarburg) Zweitgutachter: Prof. Dr. Stefan Ivanov (University of Sofia)
Eingereicht am 31.07.2014 M¨undliche Pr¨ufungam 17.10.2014 Marburg, 2014
Geometric structures and special spinor fields
Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.)
am Fachbereich der Mathematik und Informatik Philipps-Universit¨atMarburg (Hochschulkennziffer 1180)
von Dipl. Math. Jos H¨oll
geboren am 13.05.1984 in Herrenberg
Erstgutachter: Prof. Dr. habil Ilka Agricola (Universit¨atMarburg) Zweitgutachter: Prof. Dr. Stefan Ivanov (University of Sofia)
Eingereicht am 31.07.2014 M¨undliche Pr¨ufungam 17.10.2014 Marburg, 2014
iii
Deutsche Zusammenfassung
Gegen Ende der 1920er Jahre legte Paul Dirac den Grundstein f¨urdie Entwicklung von Spinoren und dem nat¨urlicher Weise auf ihnen operierenden Differentialoperator, dem Dirac-Operator ([Di28a, Di28b]). Seit diesem Zeitpunkt spielen Spinoren, die spezielle Differentialgleichungen erf¨ulleneine große Rolle in Physik und Mathematik. Es stellte sich heraus, dass spezielle Spinoren nur auf bestimmten Typen von Mannigfaltigkeiten existieren k¨onnen.Es wurden Korresponden- zen zwischen Differenzialgleichungen f¨urSpinoren und Typen von Mannigfaltigkeiten entdeckt. Einige wichtige Zusammenh¨angesind in der folgenden Liste aufgef¨uhrt.
• Parallele Spinoren existieren nur auf Ricci-flachen R¨aumen. • Der Index des Dirac-Operators ist gleich dem (rein topologischen) Aˆ Geschlecht [AS62].
• Parallele Spinoren erfordern Holonomie Sp(n), SU(n), G2 und Spin(7) [Wa89]. • Korrespondenzen zwischen Killing-Spinoren und geometrischen Strukturen [FK89, FK90, Gr90].
Wir werden die im letzten Punkt genannten Killing-Spinoren verallgemeinern, wie es beispiels- weise schon in [FK01, BGM05] getan wurde, um damit neue Zusammenh¨angemit geometrischen Strukturen herzustellen. So korrespondieren zum Beispiel in Dimension 6 die generalisierten Killing-Spinoren, die f¨urden Levi-Civita Zusammenhang ∇ mit einem symmetrischen Endomor- phismus S durch die Gleichung
∇X ϕ = S(X) · ϕ beschrieben werden, zu halb-flachen SU(3)-Strukturen. Ebenso ergeben sich Korrespondenzen zum Kern des Dirac-Operators, wie sie in der Physik von großem Interesse sind; in [CCD03] wurden beispielsweise die Restriktionen f¨ureben diese Korrespondenz bereits betrachtet.
Im ersten Kapitel werden die geometrischen Strukturen eingef¨uhrt,die in dieser Arbeit behandelt werden und die oben genannten Korrespondenzen zu Spinoren untersucht. Im zweiten Kapitel wenden wir uns der Hyperfl¨achentheorie zu. Ein Spinor ϕ auf einer Mannigfaltigkeit M¯ mit Hyperfl¨ache M ⊂ M¯ erf¨ulltf¨urdie zugeh¨origenLevi-Civita Zusammenh¨ange ∇ und ∇¯ die Gleichung 1 ∇¯ ϕ = ∇ ϕ − W (X) · ϕ, (1) X X 2 wobei W die Weingarten-Abbildung ist (z.B. [BGM05]). Dies gibt uns die M¨oglichkeit, Zusam- menh¨angevon Spinoren und durch den ersten Teil der Arbeit damit eben auch zwischen geo- metrischen Strukturen auf M und M¯ herzustellen. Die meisten geometrischen Strukturen tragen einen charakteristischen Zusammenhang ∇c mit Torsion T , der metrisch ist, die Struktur erh¨altund außerdem die gleichen Geod¨atenwie der Levi-Civita Zusammenhang ∇ besitzt. Wir werden diese Zusammnh¨angenutzen, um Korrespon- denzen von geometrischen Strukturen auf Untermannigfaltigkeiten zu schaffen, die nicht durch Spinoren gegeben sind. Generalisierte Killing-Spinoren, wie sie oben beschrieben wurden, und Killing-Spinoren mit Tor- sion, definiert durch die Gleichung ∇s · X ϕ = αX ϕ (2) f¨urden Zusammenhang ∇s = ∇ + 2sT , wobei ∇ der Levi-Civita Zusammenhang und T die charakteristische Torsion ist, sind von immer gr¨oßererBedeutung (siehe z.B. [ABBK13, FK01, BGM05]). Wir werden charakteristische Zusammenh¨angenutzen, um solche Spinoren auf M und M¯ zu untersuchen.
In Abschnitt 1 des ersten Kapitels werden metrische fast-Kontakt-Strukturen eingef¨uhrt. Hier wird an die Klassifikation solcher Strukturen erinnert und außerdem ein n¨utzliches Kriterium iv f¨urdie Existenz eines charakteristischen Zusammenhangs gegeben. Wir nutzen dieses Kri- terium, um zu zeigen, dass in der Klasse C13467 der Klassifikation von Chinea und Gonzalez ([CG90]) ein charakteristischer Zusammenhang existiert, nicht aber in C2, C5, C9, C10, C11 oder C12 (Theorem 1.3). Als n¨achstes werden in Abschnitt 2 fast-hermitsche Strukturen definiert und ebenfalls deren Klas- sifikation sowie das bekannte Kriterium ([FI02]) zur Existenz charakteristischer Zusammenh¨ange wiederholt. Im speziellen Fall der Dimension 6 werden in Abschnitt 3 SU(3)-Strukturen eingef¨uhrt.Da die Gruppe SU(3) ⊂ Spin(6) der Stabilisator eines Spinors ist, k¨onnenhier Korrespondenzen zwis- chen dem definierenden Spinor und der resultierenden Struktur geschaffen werden. So wird hier die Klassifikation der SU(3)-Strukturen in spinorielle Gleichungen ¨ubersetzt. Es werden Gr¨oßen wie der Nijenhuis-Tensor und der Dirac-Operator betrachtet. So ist beispielsweise das Kodif- ferential der K¨ahlerformdurch den Dirac-Operator D und die klassifizierende 1-Form η gegeben mittels δω(X) = 2[(Dϕ, X · j · ϕ) − η(X)].
Zudem wird eine sehr interessante Klasse von SU(3)-Zusammenh¨angenbetrachtet sowie eine spinorielle Bedingung f¨urdie Existenz eines charakteristischen Zusammenhangs entwickelt. Ahnlich¨ wie im vorherigen Abschnitt wird in Abschnitt 4 eine Korrespondenz zwischen Spinoren und G2-Strukturen in Dimension 7 geschaffen. Beispielsweise ist ein Spinor der L¨ange1 genau dann im Kern des Dirac-Operators, wenn die zugeh¨orige G2-Struktur aus der Klasse W23 ist. Wie im vorherigen Abschnitt in Dimension 6, werden auch hier an die G2-Geometrie angepasste Zusammenh¨angen¨ahererl¨autert. In Abschnitt 5 werden kurz Spin(7)-Strukturen in Dimension 8 eingef¨uhrt. Da ein Spinor in dieser Dimension jedoch nicht immer die Gruppe Spin(7) als Stabilisator besitzt, kann hier keine Korrespondenz wie in den obigen F¨allengegeben werden.
Im zweiten Kapitel liegt der Fokus auf der Hyperfl¨achentheorie. Tr¨agteine Mannigfaltigkeit M¯ mit Hyperfl¨ache M eine bestimmte G-Struktur so l¨asstsich diese in eine andere geometrische Struktur auf M ¨uberf¨uhren.Mittels der Beziehung aus Gleichung (1) und der jeweilig definieren- den Spinoren wird im Fall dim M = 6 und dim M¯ = 7 in Abschnitt 1 eine Korrespondenz zwischen G2-Strukturen und deren SU(3)-Hyperfl¨achen ausgearbeitet. Es werden außerdem verallgemein- erte Killing-Spinoren mit Torsion (eine Verallgemeinerung der Gleichung (2)) eingef¨uhrtund deren Korrespondenzen auf M und M¯ bestimmt. Wir nutzen die Kegelkonstruktion, um aus einer SU(3)-Mannigfaltigkeit eine G2-Mannigfaltigkeit eines bestimmten Typs zu konstruieren. Außerdem werden Beziehungen zwischen generalisierten Killing-Spinoren mit Torsion auf M und M¯ geschaffen. Aus dem oben im Fall der Spin(7)-Struktur erw¨ahnten Grund lassen sich in anderen Dimen- sionen als dim M = 6 und dim M¯ = 7 mittels Spinoren keine Beziehungen f¨urdie von uns betrachteten geometrischen Strukturen auf Hperfl¨achen herstellen. Daher werden in Abschnitt 2 Zusammenh¨angemit Torsion betrachtet. Diese sind im Folgenden ein wichtiges Werkzeug f¨urdie Untersuchung von Korrespondenzen zwischen geometrischen Strukturen auf M und dem Kegel M¯ , selbst wenn diese Strukturen nicht durch Spinoren gegeben sind. In [B¨a93]wird diese Konstruktion benutzt, um eine Beziehung zwischen Riemannschen Killing-Spinoren und der geometrischen Struktur zu schaffen. Wir benutzen diese Zusammenh¨angeaußerdem, um Killing-Spinoren mit Torsion auf Hyperfl¨achen zu betrachten. So wird in Abschnitt 3 auf dem Kegel einer fast-Kontakt-Struktur eine fast-hermitsche Struktur konstruiert und die jeweiligen Klassifikationen mit einander in Beziehung gesetzt. Wir zeigen beispielsweise, dass eine α-Sasaki Struktur zu einer lokal-konform-K¨ahlerStruktur in Beziehung steht oder dass die beiden Nijenhuis-Tensoren die Gleichen sind. Wir benutzen die in Abschnitt 2 geschaffenen Korrespondenzen zwischen Spinoren, um zu zeigen, dass ein Killing-Spinor mit Torsion auf (M, g) einem Spinor ϕ auf dem Kegel (M,¯ g¯) = (M × R, a2r2g + dr2), der die Bedin- v gung 1 ∇¯ c ϕ + (Xy(∂ yω) ∧ ω)ϕ = 0 X 2r r erf¨ullt,entspricht. Hier ist ω die K¨ahler-Form und ∇¯ c der charakteristische Zusammenhang auf M¯ . In Abschnitt 3.4 l¨asstsich anhand der Tatsache, dass die aus den 3 fast-Kontakt-Strukturen kon- struierten Zusammenh¨angenicht die selben sind, nicht wie in den ¨ubrigenAbschnitten verfahren und wir belassen es bei einer kurzen Betrachtung der Situation. Im Falle einer G2-Struktur auf einer 7 dimensionalen Mannigfaltigkeit M tr¨agtder Kegel eine Spin(7)-Struktur. Dieser Fall sowie die Korrespondenz der Klassifikationen und der generali- sierten Killing-Spinoren mit Torsion ist in Abschnitt 4 ausgearbeitet. Hier wird beispielsweise bewiesen, dass eine Spin(7)-Struktur der Klasse U1 (bzw. U2) auf dem Kegel eine G2 Struktur auf M induziert, die niemals aus der Klasse W34 (bzw. W13) sein kann. Einen Teil dieser Ergebnisse (im Wesentlichen sind dies die Ergebnisse des Abschnittes 1 aus Kapitel I sowie die Abschnitte 2, 3 und 4 aus Kapitel II) haben wir bereits in [AH13] publiziert.
vii
Danksagung
Zu aller erst gilt mein Dank meiner Betreuerin Ilka Agricola. Sie zeigte mir den Weg zu einer erfolgreichen Promotion und unterst¨utztemich umfassend bei deren Finanzierung. Durch die gemeinsamen Begegnungen bei Seminaren und Tagungen erm¨oglichte sie mir den intensiven Aus- tausch mit anderen Forschern, Doktoranden und Post-Docs. Vor allem aber hatte sie immer Zeit f¨urmich und meine Fragen und wies mir stets die richtige Richtung. Ich m¨ochte mich aber auch bei der Person Ilka Agricola bedanken. Sie ist der Kopf einer Arbeitsgruppe, die durch sie einen famili¨arenCharakter hat. Durch ihr Engagement und ihr ¨uber das Fachliche weit hinausreichende Interesse an jedem Einzelnen des Arbeitskreises bereichert sie nicht nur jeden fachlichen Austausch sondern ebenso die zahlreichen Gespr¨ache beispielsweise in Kaffeepausen und auf Tagungsreisen. Nach der Geburt meiner Tochter Jolanda gab sie mir die M¨oglichkeit, mich intensiv um meine Familie zu k¨ummernund unterst¨utztemich sehr vielseitig.
Ebenso m¨ochte ich mich bei Thomas Friedrich bedanken. Auch er hat mich durch zahlreiche Vortr¨age,Diskussionen und das Beantworten von Fragen mathematisch sehr viel weiter gebracht. Seine Ideen und Mitarbeit beim Verfassen der gemeinsamen Paper waren auch f¨urmeine math- ematische Entwicklung von großer Bedeutung.
Ein weiterer Dank geb¨uhrtStefan Ivanov, der sich als Zweitgutachter nicht nur die Zeit nahm, meine Arbeit zu lesen, sondern auch den weiten Weg nach Marburg nicht scheute, um an meiner Disputation teilzunehmen.
Außerdem m¨ochte ich Simon Chiossi danken, der mir in seiner Zeit an der Universit¨atMarburg eine große Hilfe und Unterst¨utzungwar. In zahlreichen Diskussionen ¨uber unser gemeinsames Paper verschaffte er mir einen tieferen Einblick und half mir Vortr¨ageund Poster zu erstellen.
An dieser Stelle m¨ochte ich der gesamten Arbeitsgruppe Differentialgeometrie und Analysis danken, in der jeder Einzelne aber auch die ganze Gruppe immer offen f¨urDiskussionen, Kaffee und eine entspannte Athmosph¨arewar. Ganz besonders gilt mein Dank hierbei Tobias Weich und Panagiotis Konstantis, mit denen die Diskussionen ergebnisreich und die Abende teils lang wurden.
Ein ganz besonderer Dank gilt meiner Familie. Ohne eine solch großartige Unterst¨utzung,so viel Halt und St¨arke, w¨arees mir nicht m¨oglich gewesen, diese Arbeit zu schreiben. Ganz besonders m¨ochte ich hierbei meiner Frau Davina danken, die mich mit ihrer Uberzeugung¨ sowie ihrem wissenschaftlichen Verst¨andnissunterst¨utzte.Vor allem aber war sie immer f¨urmich da, h¨orte mir zu, baute mich immer wieder auf und erg¨anztemich zu der Person, die letztlich diese Arbeit schreiben konnte. Nur uns gemeinsam konnte es gelingen, eine wunderbare Tochter aufzuziehen und dabei zwei wissenschaftliche Laufbahnen zu verfolgen. Auch bei meinen Eltern Monika und Martin H¨ollm¨ochte ich mich an dieser Stelle bedanken. Durch ihre Offenheit und Flexibilit¨at,vor allem aber auch ihre Sicherheit stand mir immer jeder Weg offen und durch sie habe ich gelernt, meinen auch zu gehen. Ein sicherer R¨uckhalt ist die beste Voraussetzung f¨urjedes Projekt.
Vielen Dank. viii Contents
Zusammenfassung iii Danksagung vii Introduction 1
I G structures and their characteristic connections 5 1 Metric almost contact structures ...... 6 1.1 Almost contact connections ...... 7 2 Almost hermitian structures ...... 9 3 Special almost hermitian structures in dimension 6 ...... 9 3.1 Linear Algebra in dimension 6 ...... 10 3.2 SU(3) manifolds ...... 11 3.3 Spinorial characterisation ...... 14 3.4 Adapted connections ...... 19 4 G2 geometry ...... 22 4.1 Linear algebra in dimension 7 ...... 23 4.2 G2 manifolds ...... 24 4.3 Spin formulation ...... 26 4.4 Adapted connections ...... 28 5 Spin(7) structures ...... 29
II Hypersurfaces, cones and generalized Killing spinors 31 1 SU(3) hypersurfaces in G2 manifolds ...... 32 1.1 Spin cones ...... 35 1.2 Killing spinors with torsion ...... 36 2 Connections on cones and the cone construction on spinors ...... 39 2.1 The cone construction ...... 39 2.2 The cone correspondence for spinors ...... 43 3 Almost hermitian cones over almost contact manifolds ...... 44 3.1 The classification of metric almost contact structures and the correspond- ing classification of almost hermitian structures on the cone ...... 46 3.2 Corresponding spinors on metric almost contact structures and their cones 53 3.3 Examples ...... 54 3.4 Metric almost contact 3-structures ...... 59 4 G2 structures – Spin(7) structures on the cone ...... 60 4.1 The classification of G2 structures and the corresponding classification of Spin(7) structures on the cone ...... 64 4.2 Corresponding spinors on G2 manifolds and their cones ...... 69 4.3 Examples ...... 70
Bibliography 73
ix
Introduction 1
Introduction
The idea of spinors and the Dirac operator as the natural differential operator acting on them was introduced by Paul Dirac in the late 1920’s [Di28a, Di28b]. Since then spinors played an im- portant role in mathematics and physics. Spinors fulfilling special differential equations were of great interest already to Paul Dirac. The observation that special spinors require a certain type of manifold to live on was developed. The earliest example of this fact is that the existence of a parallel spinor field on a Riemannian manifold requires the manifold to be Ricci-flat. This brings out the fact, that the existence of a solution to a differential equation imposes strong conditions in the geometry. The correspondences of manifolds carrying different geometric structures and the appendant spinors fulfilling interesting equations is one main point of this thesis. The most popular and important correspondence of special spinors is the Atiyah-Singer index theorem (see [AS62]), which states that the purely topological Aˆ genus of a compact Riemannian spin manifold is equal to the index of the Dirac operator (in [AS62] for simplicity dim ≡ 0 mod 8 is assumed). Another milestone was the list of Berger (see [Be55, Si62]). He determined the Ricci-flat Rie- mannian holonomy groups, which thus are candidates for manifolds with parallel spinors. The theorem of Wang in 1989 (see [Wa89]) shows us, that these groups indeed appear. He proved that a complete simply connected irreducible non-flat Riemannian spin manifold carries a parallel spinor if and only if its Riemannian holonomy is • Sp(m) in dimension 4m, • SU(m) in dimension 2m,
• G2 in dimension 7 or • Spin(7) in dimension 8. We define a G structure to be a reduction of the frame bundle of (M, g) to a G bundle. M is then a so called G manifold. Fix a group G ⊂ SO(n). Then the classification of G structures is based on the following con- cept of intrinsic torsion. Given a G structure, the Levi-Civita connection one form has values in the corresponding Lie algebra so(n) = g ⊕ m, where m is the orthogonal complement of the Lie algebra g of G in so(n). The m part of this one form is the so called intrinsic torsion and the space m splits in irreducible representations m = m1 ⊕ .. ⊕ mk under G. If only the ml part is non-zero, the structure is said to be of class ml for some l. Popular classes have their own names. For example in dimension 6 and for G = U(3) we have nearly K¨ahlerstructures, almost K¨ahlerstructures and many others. Important G2 structures in dimension 7 are for example nearly parallel or cocalibrated. If the Levi-Civita one form takes values in g, a structure is said to be integrable. We are interested in the non-integrable case. Then the (non-zero) intrinsic torsion can be used not only for classification but to construct a connection adapted to the G structure. This connection is an important tool in the investigation of G structures as described later. With the list above this gives us a correspondence of integrable SU(n) (respectively Sp(n), Spin(7) or G2) structures and parallel spinors.
Other correspondences between spinors and geometric structures came up in terms of Killing spinors. On a Riemannian spin manifold, a spinor ϕ is said to be a Riemannian Killing spinor if it satisfies ∇X ϕ = αX · ϕ for the Levi-Civita connection ∇, the Clifford multiplication · and some constant α ∈ C. Killing spinors are geometrically interesting as they realize the limiting case of the lower bound for the eigenvalue of the Dirac operator (see [Fr80]). Again, the existence of a spinor satisfying such an equation strongly restricts the geometry. There exists a real (α ∈ R) Killing spinor on a n-dimensional manifold if it carries • an Einstein Sasaki structure in dimension n = 5 ([FK89]), 2 Introduction
• a nearly K¨ahlerstructure in dimension n = 6 ([Gr90]),
• a nearly parallel G2 structure in dimension n = 7 ([FK90]). In dimension 8, real Killing spinors exist only on the sphere ([Hi86], [BFGK91] on page 123). As this is a great restriction to the geometry (for example in dimension 6, there are only some examples known, see [Gr90, FG85]), generalizations of the Killing equation become more and more interesting (see [ABBK13, FK01, BGM05] and others), which also has to do with the following fact. A spinor does not only define a geometric structure if it is a parallel spinor or a Killing spinor. In dimensions 6 and 7, the stabilizer of a spinor is SU(3) respectively G2 and we are able to translate geometric data and classifications from structures to spinors and vice versa. In sections 3 and 4 of Chapter I we will introduce the classifications of SU(3) and G2 structures as they were developed in [CS02] and [FG82] and describe them with spinorial equations. For example we will see, that to any spinor ϕ of length one in dimension 6 (resp. ϕ¯ in dimension 7) there always exists a one form η and an endomorphism S (resp. an endomorphism S¯) such that ¯ ¯ ∇X ϕ = η(X)j · ϕ + S(X) · ϕ (resp. ∇X ϕ = S¯(X) · ϕ), (3) where j = e1 · .. · e6 for any local basis ei. If η = 0 and S is symmetric (resp. if S¯ is symmetric) then ϕ (resp. ϕ¯) is called a generalized Killing spinor (see [FK01] and [BGM05]) and corresponds to a half flat SU(3) structure in dimension 6 and a cocalibrated G2 structure in dimension 7. If in addition S (resp. S¯) is a multiple of the identity this reduces to the Killing equation. Also we see, that there are interesting correspondences to the Dirac operator. An SU(3) structure of type χ22345¯ with a certain restriction on the χ45 part corresponds to a spinor of length one in the kernel of the Dirac operator (see Theorem 3.9). Such equations involving the Dirac operator are also interesting in physics. The restrictions mentioned already came up in the work of Cardoso and others (see [CCD03]).
In hypersurface theory, generalized Killing spinors play an important role (see for example [BGM05]), since on a hypersurface M in M¯ with Levi-Civita connections ∇ and ∇¯ a spinor ϕ satisfies 1 ∇¯ ϕ = ∇ ϕ − W (X) · ϕ, (4) X X 2 where W is the symmetric Weingarten tensor. If dim M = 6 and dim(M¯ ) = 7, we are able to look at a spinor defining an SU(3) structure on M and a G2 structure on M¯ . Using Equation (3), the classification of M and M¯ can then be given in terms of each other.
Many investigations of the correspondence between spinors and geometric structures are done in other dimensions then 6 and 7 as well (see for example [Iv04] for dimension 8 or [FI03] for dimension 5) but there is not always an applicable correspondence. So, to compare classifications on M and M¯ of structures not given by a spinor we need a different tool. Correspondences of Killing spinors on a manifold M and parallel spinors its cone M¯ were first recognized by Bryant ([Br87]) in some examples. In [B¨a93]B¨artranslated the existence of Killing spinors on a manifold M to the classification of parallel spinors by Wang on its cone M¯ . We will generalize this construction using connections with torsion to compare the classification on M of almost contact structures (respectively G2 structures) and the classification on M¯ of almost hermitian structures (respectively Spin(7) structures). If the projection of the intrinsic torsion to the 3-forms is non-zero, it defines the skew symmetric torsion of a metric connection preserving the geometric structure (see [FI02]), which is typically unique (see [AFH13] for the most general case) and thus is called characteristic connection. This connection not only is metric and preserves the G structure, it also has the same geodesics as the Levi-Civita connection. The characteristic connections on M and M¯ can be used for com- parison of two geometric structures. The case of almost contact structures on M and almost Introduction 3 hermitian structures on M¯ was already discussed in physics, see [HTY12], in a less general setting.
Another generalization of Killing spinors is constructed using the characteristic connection. A spinor ϕ is said to be a Killing spinor with torsion, if it satisfies
∇X ϕ + s(XyT ) · ϕ = αX · ϕ (5) for some s ∈ R, where T is the characteristic torsion. Killing spinors with torsion became in- n−1 teresting in the last years, since for example for s = 4(n−3) they realize the equality case of the eigenvalue estimation of the Dirac operator with torsion (which in some cases is also known as the cubic Dirac operator or the Dolbeault operator), see [ABBK13]. Also, much more examples can be constructed, since the restriction to the geometry given by the existance of a Killing spinor with torsion is not as strong as the restriction given by a Riemannian Killing spinor. This richness implies, that a classification is not possible. Using Equations (5) and (4) we are able to give correspondences of spinors satisfying generalized Killing equations on M and its cone, or in some cases even correspondences of spinors on a general hypersurface M and the ones on its ambient space.
In the first Chapter we will introduce the geometric structures which will be used in this thesis. We will start with metric almost contact structures in Section 1. We cite the classification of such, given by Chinea and Gonzalez in [CG90] and give a useful criterion of the existence of a characteristic connection in Section 1.1. We use this criterion to see that for an almost contact manifold there exists a characteristic connection if it is of type C13467 but not, if it is of pure type C2, C5, C9, C10, C11 or C12 (Theorem 1.3). We shortly introduce almost hermitian structures in Section 2. The criterion for the existence of a characteristic connection in this case is already known (see [FI02]). As mentioned before, special almost hermitian structures in dimension 6 are given as the stabilizer of a spinor. In Section 3 we will describe this correspondence in detail (see Lemma 3.1). To understand this concept we will shortly introduce the corresponding spin linear algebra (Section 3.1) to then give a spinorial description of the intrinsic torsion and the classification of SU(3) structures (Sections 3.2 and 3.3). In addition to the Dirac Operator as mentioned above, we calculate the Nijenhuis tensor in terms of the defining spinor (see Lemma 3.14). We will get useful equations as the following. For the Dirac operator D and the real inner product ( , ) of spinors the codifferential of the K¨ahlerform ω in terms of the defining spinor ϕ is given by (see Lemma 3.8) δω(X) = 2[(Dϕ, X · j · ϕ) − η(X)], where η is the intrinsic one form. In terms of spinors the intrinsic torsion can easily be used to define certain SU(3) connections, which we will introduce in Section 3.4. Here are given some tools to handle the characteristic connection and to show that an SU(3) manifold carries a char- acteristic connection if it is of type χ11345¯ with a certain restriction on the 4 and 5 part of the intrinsic torsion (Theorem 3.22). In Section 4 the same is done for the structure group G2 in dimension 7. The correspondence of G2 structures and spinors is given in Lemma 4.1. For the classification in terms of spinors in Section 4.3 we calculate correspondences of the following sort. A spinor ϕ of constant length is in the kernel of the Dirac operator if and only if it defines a G2 structure of type W23. As in the SU(3) case in Section 4.4 we give a description of the torsion for the characteristic connection in terms of the defining spinor (see Theorem 4.14). In dimension 8 a spinor does not always have stabilizer Spin(7), so we were not able to give correspondences as in the 6 and 7 dimensional case. Here, there always exists a characteristic connection and so we only shortly introduce Spin(7) structures as they will be used in this thesis.
In Chapter II we concentrate on hypersurface theory and the special case of a cone construction. 7 As demonstrated in Equation (4) a spinor on a manifold M¯ defining a G2 structure can be 4 Introduction viewed as a spinor on a hypersurface M 6 inducing an SU(3) structure. The classification of both can be expressed in terms of each other as we will see in Theorems 1.4 and 1.5 of Section 1. The special case of a (twisted) cone M¯ over an SU(3) manifold is considered in Section 1.1. With this tool we are able to construct G2 structures of different types, starting with a certain SU(3) manifold. Killing spinors with torsion are the topic of Section 1.2 as they correspond to certain spinors on the cone or a more general ambient space. This correspondence is worked out in Theorem 1.8. For hypersurfaces of other dimension then 6, interesting G structures are not always given by a spinor, so we need another tool as described above and have to restrict ourselves to the case of a (twisted) cone. In Section 2.1 we introduce the construction of a twisted cone as it was done in a less general case by B¨arin [B¨a93].We make extensive use of characteristic connections and connections of the form described in Equation (5) for some T . Starting with a Riemannian manifold M with characteristic connection ∇, in Lemma 2.4 we prove, that a spinor on the cone being parallel for a certain connection with torsion corresponds a ∇-Killing spinor on M. Section 2 provides the tools we will apply in the next two sections to certain dimensions and G structures. In Section 3 we concentrate on a manifold M with almost contact structure. A twisted cone over such a manifold carries an almost hermitian structure (Theorem 3.2) and with the tools described above, we are able to compare the two classifications (Section 3.1). We see for example that an α-Sasaki structure corresponds to a locally conformally K¨ahlerstructure on the cone (Theorem 3.12). In Lemma 3.7 we additionally prove, that the two Nijenhuis tensors are basically the same. We also apply the spinorial correspondences of Section 2.2 to this case to get interesting spinorial equations in terms of the data of the geometric structure. For example we get a one to one correspondence between Killing spinors with torsion on (M, g) and spinors ϕ on the cone (M,¯ g¯) = (M × R, a2r2g + dr2) for some fixed a > 1 satisfying 1 ∇¯ c ϕ + (Xy(∂ yω) ∧ ω)ϕ = 0, X 2r r where ω is the K¨ahlerform and ∇¯ c is the characteristic connection on M¯ . For examples of this case see Section 3.3. We also take a quick look on metric almost contact 3-structures (the more general case of a 3-Sasakian structure) in Section 3.4. But since the connections we construct to each of the three almost contact structures do not coincide, we shall only make a few comments here. However, in dimension 7, 3-Sasakian manifolds carry a cocalibrated G2 structure, which then has a characteristic connection ([AF10]). This case is discussed in terms of G2 structures in Section 4, Example 4.18. We continue with the investigation in dimension 7. We look at G2 structures and their corre- sponding Spin(7) structures on the cone to compare the two classifications in Section 4.1. In Theorem 4.13 we show, that a Spin(7) structure of type U1 on the cone induces a G2 structure, which is never of type W34 and that a structure of type U2 leads to a G2 structure, which cannot be of type W13. Again we calculate correspondences of spinors on a G2 manifold and spinors on its Spin(7) cone in terms of the geometric data to give interesting examples in Section 4.3. Some of this results (mainly the results from Section 1 of Chapter I and Sections 2, 3 and 4 from Chapter II) we already published in [AH13]. Chapter I
G structures and their characteristic connections
Let (M, g) be an oriented Riemannian manifold with Levi-Civita connection ∇g witch connection 1-form Z. By definition, a G structure on M is a reduction of the frame bundle of M to some closed subgroup G ⊂ SO(n). For the classification of such structures we consider the connection 1-form Z with values in so(n). We decompose
so(n) = g ⊕ m, where g is the Lie algebra of G and consider the corresponding splitting Z = Z∗ + Γ. Then Γ is called intrinsic torsion of the G structure. Again we decompose m into irreducible representations of G giving the classes we use for classification. In some cases we will look at the connection Z∗ with linear connection ∇n and calculate its connection type given by the decomposition of the space of all metric connections TM ⊕ Λ3(TM) ⊕ T where the parts are called vectorial, skew symmetric and cyclic traceless. See Section 3.4 for more details on this decomposition.
If M admits a metric connection ∇c with skew symmetric torsion T c preserving the G structure, it will be called a characteristic connection. This is a metric connection which is adapted to the structure (rather then ∇g) but still has the same geodesics then ∇g. In sections 3 and 4 of this chapter and in Section 1 of Chapter II we will mostly work with the Levi-Civita connection and thus shorten ∇g to ∇, while in the other sections the characteristic connection is used more frequently and thus often ∇c will be shortened to ∇.
The following result proves the uniqueness of the characteristic connection in many geometric situations:
Theorem 0.1 ([AFH13, Thm 2.1.]). Let G ( SO(n) be a connected Lie subgroup acting ir- reducibly on Rn, and assume that G does not act on Rn by its adjoint representation. Then the characteristic connection of a G structure on a Riemannian manifold (M, g) is, if existent, unique.
This applies, for example, to almost hermitian structures (U(n) ⊂ SO(2n)), G2 structures in dimension 7 and Spin(7) structures in dimension 8 (but not to metric almost contact structures). We will now introduce the G structures considered in this thesis.
5 6 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
1 Metric almost contact structures
Let M be a n = 2k + 1 dimensional manifold. Given a Riemannian metric g, a (1,1)-tensor ψ : TM → TM, a 1-form η with dual vector field ξ of length one, and the (2, 0)-tensor F defined by F (v, w) := g(v, ψ(w)), we call (M, g, ψ, η) a metric almost contact structure if
ψ2 = −id + η ⊗ ξ and g(ψv, ψw) = g(v, w) − η(v)η(w).
In [Bl02, Thm 4.1.D], D. Blair shows that ψ(ξ) = 0 and η ◦ ψ = 0. Since
g(v, ψ(w)) = g(ψ(v), ψ2(w)) + η(v)η(ψ(w)) = g(ψ(v), −w + η(w)ξ) = −g(ψ(v), w), for all v, w ∈ TM, F is actually a 2-form. In terms of the Levi-Civita connection ∇g on M, the Nijenhuis tensor of a metric almost contact structure is defined by ∇g − ∇g ∇g − ∇g N(X,Y,Z) := g(( X ψ)(ψ(Y )) ( Y ψ)(ψ(X)) + ( ψ(X)ψ)(Y ) ( ψ(Y )ψ)(X),Z) ∇g − ∇g + η(X)g( Y ξ, Z) η(Y )g( X ξ, Z). The classification of metric almost contact structures is relatively involved. For future reference, we recall in the following table the exact definition of the different classes of of n-dimensional metric almost contact manifolds given by Chinea and Gonzalez [CG90].
class defining relation
C ∇g ∇g 1 ( X F )(Y,Z) = 0, η = 0 g C2 dF = ∇ η = 0 C ∇g − ∇g 3 ( X F )(Y,Z) ( ψX F )(ψY, Z) = 0 C ∇g − 1 − 4 ( X F )(Y,Z) = n−3 [g(ψX, ψY )δF (Z) g(ψX, ψZ)δF (Y ) −F (X,Y )δF (ψZ) + F (X, Z, δF (ψY )], δF (ξ) = 0 C ∇g 1 − 5 ( X F )(Y,Z) = n−1 [F (X,Z)η(Y ) F (X,Y )η(Z)]δη C ∇g 1 − 6 ( X F )(Y,Z) = n−1 [g(X,Z)η(Y ) g(X,Y )η(Z)]δF (ξ) C ∇g ∇g ∇g 7 ( X F )(Y,Z) = η(Z)( Y η)(ψX) + η(Y )( ψX η)(Z), δF = 0 C ∇g − ∇g ∇g 8 ( X F )(Y,Z) = η(Z)( Y η)(ψX) + η(Y )( ψX η)(Z), δη = 0 C ∇g ∇g − ∇g 9 ( X F )(Y,Z) = η(Z)( Y η)(ψX) η(Y )( ψX η)(Z) C ∇g − ∇g − ∇g 10 ( X F )(Y,Z) = η(Z)( Y η)(ψX) η(Y )( ψX η)(Z) C ∇g − ∇g 11 ( X F )(Y,Z) = η(X)( ξ F )(ψY, ψZ) C ∇g ∇g − ∇g 12 ( X F )(Y,Z) = η(X)η(Z)( ξ η)(ψY ) η(X)η(Y )( ξ η)(ψZ)
The most important classes are
•C3 ⊕ .. ⊕ C8, the normal structures characterized by N = 0,
•C6 ⊕ C7, the quasi Sasaki structures: normal structures satisfying dF = 0,
•C6, the α-Sasaki structures: normal structures with αF = dη for some constant α, • Sasaki structures: α-Sasaki structures with δF (ξ) = n − 1. 1. METRIC ALMOST CONTACT STRUCTURES 7
Other classifications we will not consider here are formulated in terms of the Nijenhuis tensor or by considering the direct (not the twisted) product M × R ([CM92] and [Ou85]). It turns out ∇g that the tensor α(X,Y,Z) := ( X F )(Y,Z) will be a useful tool for the investigation of metric almost contact structures. It satisfies the general formula
α(X,Y,Z) = −α(X,Z,Y ) = −α(X, ψY, ψZ) + η(Y )α(X, ξ, Z) + η(Z)α(X, Y, ξ). (I.1)
This implies
α(X, Y, ψY ) = −α(X, ψY, ψ2Y ) + η(Y )α(X, ξ, ψY ) = −α(X, Y, ψY ) + 2η(Y )α(X, ξ, ψY ), so we have α(X, Y, ψY ) = η(Y )α(X, ξ, ψY ). (I.2)
1.1 Almost contact connections A metric almost contact structure admits a characteristic connection if and only if its Nijenhuis tensor is skew symmetric and ξ is a Killing vector field, and then it is unique [FI02, Thm 8.2]. If it exists, its torsion tensor is given by
T = η ∧ dη + dF ψ + N − η ∧ (ξyN), where dF ψ := dF ◦ ψ. We shall now prove a useful criterion for the existence of a characteristic connection. Lemma 1.1. A metric almost contact manifold (M, g, ψ, η) admits a characteristic connection if and only if ∇g ∇g ( Y F )(Y, ψX) + ( ψY F )(Y,X) = 0. Proof. There exists a characteristic connection if and only if the Nijenhuis tensor N is skew symmetric and ξ is a Killing vector field. Since we have ∇g − ∇g ∇g ∇g g( Y ξ, Z) = F ( Y ξ, ψZ) = ( Y F )(ξ, ψZ) = ( Y η)(Z) ∇g ∇g and ( X F )(Z,Y ) = g(( X ψ)Y,Z), the Nijenhuis tensor on M may be written as N(X,Y,Z) = α(X, Z, ψY ) − α(Y, Z, ψX) + α(ψX, Z, Y ) − α(ψY, Z, X) + η(X)α(Y, ξ, ψZ) − η(Y )α(X, ξ, ψZ).
Thus N is skew symmetric if
0 = N(X,Y,Y ) = α(X, Y, ψY )−α(Y, Y, ψX)−α(ψY, Y, X)+η(X)α(Y, ξ, ψY )−η(Y )α(X, ξ, ψY ).
With equation (I.2), N is skew symmetric if and only if
0 = −α(Y, Y, ψX) − α(ψY, Y, X) + η(X)α(Y, ξ, ψY ). (I.3) ∇g ∇g ξ is a Killing vector field if 0 = g( X ξ, Y ) + g( Y ξ, X) = α(X, ξ, ψY ) + α(Y, ξ, ψX), and this is satisfied if and only if α(Y, ξ, ψY ) = 0. Together with condition (I.3) we obtain the condition
0 = α(Y, Y, ψX) + α(ψY, Y, X).
To see that this is also sufficient, set X = ξ. We define Definition 1.2. A metric almost contact manifold admitting a characteristic connection is called a metric almost contact manifold with torsion. 8 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
With the above lemma we can easily prove Theorem 1.3. Consider a metric almost contact manifold (M, g, ψ, η). If it is of class
1. C1 ⊕ C3 ⊕ C4 ⊕ C6 ⊕ C7, there exists a characteristic connection.
2. C2, C5, C9, C10, C11 or C12 there is no characteristic connection.
3. C8 there exists a characteristic connection if and only if ξ is a Killing vector field. Proof. We check the different cases: In C1 we have α(X,X,Y ) = α(X, Z, ξ) = 0 and we thus get α(Y, Y, ψX) + α(ψY, Y, X) = 0. For a structure given by α in the class C2 we have α(X,Y,Z) + α(Y,Z,X) + α(Z,X,Y ) = α(X, Y, ξ) = 0, and equation (I.2) yields α(Y, Y, ψX) + α(ψY, Y, X) = α(Y, Y, ψX) − α(Y, X, ψY ) − α(X, ψY, Y ) (I.1) = α(Y, Y, ψX) + α(Y, ψY, X) = −α(Y, ψY, ψ2X) + α(Y, ψY, X) = 2α(Y, Y, ψX). Thus the condition α(Y, Y, ψX)+α(ψY, Y, X) = 0 implies 0 = α(Y, Y, ψ2X) = −α(Y,Y,X) since α(Y, Y, ξ) = 0. Therefore α has to be also of class C1, which implies α = 0. In C3 we have α(X,Y,Z) = α(ψX, ψY, Z) and get α(Y, Y, ψX) + α(ψY, Y, X) = α(Y, Y, ψX) − α(ψY, X, Y ) = α(Y, Y, ψX) − α(ψ2Y, ψX, Y ) = α(Y, Y, ψX) + α(Y, ψX, Y ) = 0 since α(ξ, X, Y ) = 0 in C1 ⊕ ... ⊕ C10. A structure is of class C3 ⊕ ... ⊕ C8 if and only if N = 0 thus we just have to check the condition α(Y, ξ, ψY ) = 0, which is satisfied in C4 and C6. C δη − 5 is given by the condition α(X,Y,Z) = n−1 (F (X,Z)η(Y ) F (X,Y )η(Z)) such that the condition α(Y, ξ, ψY ) = 0 implies δη = 0 and thus α = 0. For (c, b) = (1, −1) in C7,(c, b) = (−1, −1) in C8,(c, b) = (1, 1) in C9 and (c, b) = (−1, 1) in C10 we have α(X,Y,Z) = cη(Z)α(Y, X, ξ) + bη(Y )α(ψX, ψZ, ξ) and get α(X, Y, ξ) = cα(Y, X, ξ) and α(X, ψY, ξ) = bα(X, ψY, ξ), implying (1−cb)α(Y, ψY, ξ) = 0. Thus in C7 and C10 the vector field ξ is Killing. Since in C7 we have N = 0, we have a characteristic connection here. In C8 we have a characteristic connection if and only if ξ is Killing. In C9 and C10 we have b = 1 and thus α(Y, Y, ψX) + α(ψY, Y, X) = − η(Y )α(ψY, X, ξ) + cη(X)α(Y, ψY, ξ) − η(Y )α(Y, ψX, ξ) = − 2η(Y )α(ψY, X, ξ) + cη(X)α(Y, ψY, ξ). For X = ξ the condition α(Y, Y, ψX) + α(ψY, Y, X) = 0 implies α(Y, ψY, ξ) = 0 and thus we have 0 = α(ψY, X, ξ) and also 0 = α(ψ2Y, X, ξ) = −α(Y, X, ξ) since α(ξ, X, Y ) = 0. So we have already α = 0. C11 is given by the condition α(X,Y,Z) = −η(X)α(ξ, ψY, ψZ) and thus with α(ξ, ξ, X) = 0 we get α(Y, Y, ψX) + α(ψY, Y, X) = η(Y )α(ξ, ψY, X). Because α(ξ, ψY, X) = 0 already implies α(ξ, Y, X) = 0, we obtain in this case immediately α = 0. In C12 we have α(X,Y,Z) = η(X)η(Y )α(ξ, ξ, Z)+η(X)η(Z)α(ξ, Y, ξ) and thus 0 = α(Y, Y, ψX)+ α(ψY, Y, X) = η(Y )2α(ξ, ξ, ψX) gives us α = 0. 2. ALMOST HERMITIAN STRUCTURES 9
Remark 1.4. The conditions for a metric almost contact structure to admit a characteristic connection in Theorem 1.3 are sufficient but not necessary. In [Pu12] C. Puhle proves that in the case n = 5, there are structures of class C10 ⊕ C11 (in his class W4) carrying a characteristic connection. Thus a structure with characteristic connection is never of pure class C10 nor of class C11, but it can be of mixed class C10 ⊕ C11. But more detailed descriptions are possible in some cases. For example, if we set Y = ξ, the equation 0 = α(Y, Y, ψX) + α(ψY, Y, X) immediately implies that a structure with characteristic connection is of class C1 ⊕ ... ⊕ C11.
2 Almost hermitian structures
Let (M, g) be a 2m-dimensional Riemannian manifold equipped with a (1, 1)-tensor
2 J : TM → TM with J = −IdTM , and g(JX,JY ) = g(X,Y ).
We define a 2-form ω(X,Y ) := g(X,JY ). Then (M, g, J, ω) is called an almost hermitian man- ifold. In terms of the Levi-Civita connection ∇g on M, the Nijenhuis tensor of M is defined to be
∇g − ∇g ∇g − ∇g N(X,Y,Z) = g(( X J)(JY ),Z) g(( Y J)(JX),Z) + g(( JX J)(Y ),Z) g(( JY J)(X),Z). Almost hermitian structures were classified by Gray and Hervella in [GH80] into four classes χ1 ⊕ χ2 ⊕ χ3 ⊕ χ4, which we recall in the following table.
name class defining relation
∇g nearly K¨ahler χ1 ( X J)X = 0
almost K¨ahler χ2 dω = 0
balanced χ3 N = 0 and δω = 0 ∇¯ g −1 − locally conformally K¨ahler χ4 ( X ω)(Y,Z) = n−1 [g(X,Y )δω(Z) g(X,Z)δω(Y ) −g(X,JY )δω(JZ) + g(X,JZ)δω(JY )]
An almost hermitian manifold admits a characteristic connection if and only if it is of class χ1 ⊕ χ3 ⊕ χ4 [FI02] and it is always unique (either by explicit computation as in [FI02] or by c the general Theorem 0.1). Due to the fact that in class χ1 ⊕ χ3 ⊕ χ4 we have ∇ ω = 0 such manifolds are sometimes called K¨ahlermanifolds with torsion, although they are evidently not K¨ahlerian.Their characteristic torsion is given by (see for example [Ag06])
T = N + dωJ ,
J where dω := dω ◦ J. For a nearly K¨ahlermanifold (class χ1), this connection was first in- troduced and investigated by A. Gray; on hermitian manifolds (N = 0, i. e. class χ3 ⊕ χ4) it is sometimes called the Bismut connection [Bi89]. Almost hermitian manifolds of class χ4 are locally conformally K¨ahlermanifolds.
3 Special almost hermitian structures in dimension 6
Let (M, g) be a 6 dimensional Riemannian manifold with a 3 form ψ such that in some local basis e1, ..e6 the form ψ reads as
ψ = e1 ∧ e3 ∧ e6 − e1 ∧ e4 ∧ e6 − e2 ∧ e3 ∧ e6 − e3 ∧ e4 ∧ e5. 10 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
Then ψ is a complex determinant corresponding to the almost hermitian structure
ω = e1 ∧ e3 + e1 ∧ e4 + e5 ∧ e6 and its stabilizer is SU(3) ⊂ SO(6). Such a structure can rather be defined via spinors: ∼ The group SU(3) ⊂ SO(6) is simply connected and lifts to SU(3) ⊂ Spin(6) = SU(4). Thus an SU(3) structure can be described in terms of a stabilizing spinor. This gives a new view on SU(3) structures, which will be described in the following. This viewpoint is very useful in the hypersurface theory considered in Chapter II. To see this correspondence correctly, we need to look at the linear algebra of dimension 6.
3.1 Linear Algebra in dimension 6
6 + − We consider R and the corresponding Spin(6) representation ∆6 = ∆ ⊕ ∆ . In ∆6 exists a 6 real structure α : ∆6 → ∆6 with the following properties for all ϕ, ϕ1, ϕ2 ∈ ∆6, X ∈ R and the hermitian scalar product ⟨ , ⟩ on ∆6 (see [Fr00], Section 1.7). • α is real linear, • α(iϕ) = −iα(ϕ),
2 • α = Id∆, • α interchanges ∆+ and ∆−, α : ∆± → ∆∓, • α interchanges with the Clifford multiplication α(Xϕ) = Xα(ϕ) and
•⟨α(ϕ1), ϕ2⟩ = ⟨α(ϕ2), ϕ1⟩.
Let ∆ := {ϕ ∈ ∆6 | α(ϕ) = ϕ} be the real Spin(6) representation with real scalar product ( .,. ). Define in the Clifford algebra of R6 the element
j := e1 · ... · e6. As an endomorphism of ∆, j satisfies j2 = −Id, anti commutes with the Clifford multiplication and (j(ϕ), ϕ) = −(ϕ, j(ϕ)). Thus j defines a Spin(6) invariant complex structure and in fact Spin(6) is isomorphic to SU(4). Given a one dimensional subspace V = span(ϕ) ⊂ ∆ for any ϕ of length one we get dimR{Y ϕ | Y ∈ R6} = 6 since the Clifford multiplication with an element Y ∈ R6 is an isomorphism. ϕ and j(ϕ) are orthogonal to X · ϕ since
(X · ϕ, j(ϕ)) = −(ϕ, X · j(ϕ)) = (ϕ, j(X · ϕ)) = −(j(ϕ),X · ϕ) = −(X · ϕ, j(ϕ)).
This gives us the splitting
∆ = Rϕ ⊕ Rj(ϕ) ⊕ {X · ϕ | X ∈ R6}. (I.4)
6 6 Thus we can define an orthogonal complex structure Jϕ : R → R by
Jϕ(X) · ϕ = j(X · ϕ). 2 − R6 We immediately get Jϕ = IdR6 and g(JX,JY ) = g(X,Y ) for the standard metric g on . Clearly Jϕ does not depend on the choice of ϕ ∈ V . The defining 2-form of the hermitian structure is given by
ω(X,Y ) := g(X,JY ) = (Xϕ, JY ϕ) = −(Xϕ, Y j(ϕ)) = (ϕ, XY j(ϕ)) = −(j(ϕ), XY ϕ). 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 11
We can also define the 3-form ψϕ via ϕ:
ψϕ(X,Y,Z) := −(X · Y · Z · ϕ , ϕ) . (I.5)
ψϕ is invariant under J
ψϕ(JX,Y,Z) = ψϕ(X,JY,Z) = ψϕ(X,Y,JZ) and thus a complex determinant. This reduces SO(6) to SU(3).
Since SU(3) is simply connected, if conversely given a reduction SU(3) ⊂ SO(6) we get a lift ∼ ∼ SU(3) ⊂ Spin(6) = SU(4) fixing a complex one dimensional subspace V ⊂ ∆ = C4. Taking definition (I.5) together with ||ϕ|| = 1 as a condition, we thus get a spinor ϕ ∈ V , which is unique up to ±1. Lemma 3.1. There is a one to one correspondence between • complex structures with a complex determinant on R6, • reductions of SO(6) to SU(3), • reductions of Spin(6) to SU(3), • real one dimensional subspaces of the real Spin(6) representation ∆. Thus the space of special hermitian structures on R6 is given by
RP(∆) = RP(7) = SO(6)/SU(3), where RP(∆) is the real projective space over the vector space ∆.
We summarize some formulas expressing the action of the 2- and 3-form Jϕ and ψϕ Lemma 3.2.
∗ ∗ ψϕ · ϕ = −4 · ϕ, ψϕ · j(ϕ) = 4 · j(ϕ), ψϕ · ϕ = 0 if ϕ ⊥ ϕ, j(ϕ),
6 (X ψϕ) · ϕ = 2 X · ϕ X ∈ R ,Jϕ · ϕ = 3 j(ϕ),Jϕ · j(ϕ) = −3 ϕ.
3.2 SU(3) manifolds An SU(3) manifold (M 6, g, ϕ) is a Riemannian spin manifold equipped with a global spinor ϕ of length one. We always denote its spinor bundle by Σ and its Levi-Civita connection by ∇. The induced SU(3)-structure is given by the 3-form ψϕ. Let ω = g(., J.) be the hermitian 2-form defining the corresponding U(3)-structure. We define a second 3-form by J − − ψϕ (X,Y,Z) := ψϕ(JX,JY,JZ) = ψϕ(JX,Y,Z) = (XY Zϕ, j(ϕ)). We shall recover the various SU(3)-types essentially by reinterpreting the intrinsic torsion. With the splitting (I.4) we have ∇ϕ = η ⊗ j(ϕ) + S ⊗ ϕ, for some 1-form η and a linear map S ∈ End(TM 6). Moreover we have Lemma 3.3. S and η are given by ∇ J − ∇ J ( X ω)(Y,Z) = 2ψϕ (S(X),Y,Z) and 8η(X) = ( X ψϕ )(ψϕ) for any X,Y,Z. 12 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
Proof. We immediately find η = (∇ϕ, j(ϕ)). Since j is the volume form, it is parallel under ∇ and we conclude
∇X (j(ϕ)) = j∇X ϕ = jS(X)ϕ + jη(X)j(ϕ) = −S(X)j(ϕ) − η(X)ϕ. With ω(X,Y ) = −(Xϕ, Y j(ϕ)) we get
−(∇X ω)(Y,Z) = X(Y ϕ, Zj(ϕ)) − (∇X Y ϕ, Zj(ϕ)) − (Y ϕ, ∇X Zj(ϕ))
= (∇X (Y ϕ), Zj(ϕ)) + (Y ϕ, ∇X (Zj(ϕ))) − (∇X Y ϕ, Zj(ϕ)) − (Y ϕ, ∇X Zj(ϕ))
= (Y ∇X ϕ, Zj(ϕ)) + (Y ϕ, Z∇X j(ϕ)) = (YS(X)ϕ, Zj(ϕ)) − (Y ϕ, ZS(X)j(ϕ)) − η(X)(Y ϕ, Zϕ) + η(X)(Y j(ϕ), Zj(ϕ)) = −(ZYS(X)ϕ, j(ϕ)) − (S(X)ZY ϕ, j(ϕ)) J J = ψϕ (Z,Y,S(X)) + ψϕ (S(X),Z,Y ) J − J = ψϕ (Y,S(X),Z) ψϕ (S(X),Y,Z) − J = 2ψϕ (S(X),Y,Z) Furthermore, the computation ∇ J − ∇ X (ψϕ )(ψϕ) = X(ψϕϕ, j(ϕ)) + ( X ψϕϕ, j(ϕ))
= −(ψϕ∇X ϕ, j(ϕ)) − (ψϕϕ, ∇X j(ϕ))
= −(ψϕS(X)ϕ, j(ϕ)) + (ψϕϕ, S(X)j(ϕ))
−η(X)(ψϕj(ϕ), j(ϕ)) + η(X)(ψϕϕ, ϕ)
= 2η(X)(ψϕϕ, ϕ) = −8η(X) finishes the proof. To better understand the role of the pair (S, η) we will work with the SU(3)-connection ∇n ∇ − X Y = X Y Γ(X)(Y ), given by the Levi-Civita connection ∇ minus the intrinsic torsion Γ, see [Sa89]. Decompose so(6) = su(3) ⊕ m, then Γ is a one form with values in m. We shall repeatedly use one symbol for covariant derivatives on the tangent bundle and lifted covariant derivatives on the spin bundle, hence ∇n ∗ ∇ ∗ − 1 ∗ X ϕ = X ϕ 2 Γ(X)ϕ for any spinor ϕ∗. Proposition 3.4. The intrinsic torsion of the SU(3)-structure (M 6, g, ϕ) is given by 2 Γ = Sy ψ − η ⊗ ω ϕ 3 where Sy ψϕ(X,Y,Z) := ψϕ(S(X),Y,Z). ∇n ∇ 1 Proof. The spinor ϕ is parallel for , as Stab(ϕ) = SU(3), so X ϕ = 2 Γ(X)ϕ. By Lemma 3.2 we have ωϕ = −3j(ϕ), so 1 1 ∇ ϕ = S(X)ϕ + η(X)j(ϕ) = (S(X)y ψ )ϕ − η(X)ωϕ. X 2 ϕ 3 ⊥ ⊥ Since (Xy ψϕ)(Y,JϕZ) = (Xy ψϕ)(JϕY,Z) we see that Xy ψϕ ∈ su(3) and since ω ∈ su(3) , y − 2 ⊗ the 1-form S ψϕ 3 η ω is the intrinsic torsion of the spinorial connection: Suppose Γ = Γ˜ + r 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 13 for some r with values in su(3)⊥. Then we have
Γ(˜ X)ϕ = 2∇X ϕ = Γ(X)ϕ = Γ(˜ X)ϕ + r(X)ϕ and thus r(X)ϕ = 0. Since the stabilizer of ϕ is SU(3) this implies r(X) ∈ su(3) and with the assumption r(X) ∈ su(3)⊥ we get Γ = Γ.˜ In the light of this fact we may call S the intrinsic endomorphism and η the intrinsic 1-form, for reference. The classification of the SU(3) structure is given by η and S and we consider the space TM ⊕ End(TM) of all such forms. Under SU(3) we have the decomposition
End(R6) = R · J ⊕ R · Id ⊕ su(3) ⊕ { ∈ 2 R6 | } A S0 ( ) AJ = JA ⊕ { ∈ 2 R6 | − } A S0 ( ) AJ = JA ⊕ {A ∈ Λ2(R6)|AJ = −JA} with dimensions 36 = 1 + 1 + 8 + 8 + 12 + 6. We compare those to the classes of special almost − ⊕ + ⊕ − ⊕ + ⊕ ⊕ ⊕ hermitian structures χ1 χ1 χ2 χ2 χ3 χ4 χ5 given in [CS02]. An SU(3) structure induces ⊕ ⊕ 5 5 an U(3) structure. This U(3) structure is of type χi1 .. χik for 1 i1 < .. < ik 4 in the ⊕ ⊕ ⊕ Gray-Hervella classification [GH80] if and only if the SU(3) structure is of type χi1 .. χik χ5. Thus an SU(3) structure is of type χ5 if the U(3) structure is K¨ahlerand 1-form η determines the class χ5. Comparing the dimensions of the other SU(3) modules one can identify the classes. + − We only need a closer look at χi and the χi parts for i = 1, 2. It suffices to look at Example 3.12 or at Remark 3.7. In the following we use + ⊕ − ⊕ + ⊕ − ⊕ ⊕ ⊕ Notation 3.5. From now on we will write χ112¯ 2345¯ for χ1 χ1 χ2 χ2 χ3 χ4 χ5 and + ⊕ − ⊕ denote subspaces in the obvious way, for example χ1 χ2 χ4 will be written as χ124¯ . Thus + − χ1 and χ1 will be denoted by χ1 and χ1¯. Lemma 3.6. The classes of an SU(3) structure (M 6, g, ϕ) are determined as follows.
class description dimension
χ1 S = λ · Jϕ, η = 0 1
χ1¯ S = µ · Id, η = 0 1
χ2 S ∈ su(3), η = 0 8 ∈ { ∈ 2 R6 | } χ2¯ S A S0 ( ) AJϕ = JϕA , η = 0 8 ∈ { ∈ 2 R6 | − } χ3 S A S0 ( ) AJϕ = JϕA , η = 0 12 2 6 χ4 S ∈ {A ∈ Λ (R )|AJϕ = −JϕA}, η = 0 6
χ5 S = 0, η ≠ 0 6 where λ, µ ∈ R. In particular S is symmetric and η = 0 if and only if the type is χ1¯23¯ . − ⊕ − ⊕ Remark 3.7. An SU(3) structure is half flat (of type χ1 χ2 χ3), if the manifold M is possibly a submanifold of a manifold M¯ with holonomy contained in G2 (see [CS02]). The G2 structure on M¯ thus is given by a parallel spinor. On the other hand we showed that being half flat means ∇X ϕ = S(X)ϕ with some symmetric S. Thus ϕ can possibly be lifted to a flat spinor on a manifold M¯ with Weingarten map S (see [BGM05]), defining a flat G2 structure and the two statements are the same. 14 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
3.3 Spinorial characterisation The description of SU(3)-structures in terms of ϕ is the main result of this section. To start with we will describe some geometric quantities of the SU(3) structure and their correspondence to ϕ. Denote by D the Riemannian Dirac operator.
Lemma 3.8. On an SU(3) manifold the χ4-component of the intrinsic torsion is given by
δω(X) = 2[(Dϕ, Xj(ϕ)) − η(X)].
In particular δω = 0 is equivalent to (Dϕ, Xj(ϕ)) = η(X). The Lee form is given by
θ(X) = δω ◦ J(X) = 2(Dϕ, Xϕ) − 2η ◦ J(X).
Proof. We have
(∇X ω)(Y,Z) = (ZY ∇X ϕ, j(ϕ)) + (ZY ϕ, ∇X j(ϕ)) = −2(YZ∇X ϕ, j(ϕ)) − 2g(Y,Z)η(X), leading to ∑ ∑ − ∇ ∇ δω(X) = ( ei ω)(ei,X) = ( ei ω)(X, ei) ∑i i − ∇ − = 2 ((Xei ei ϕ, j(ϕ)) g(X, ei)η(ei)) i = −2(XDϕ, j(ϕ)) − 2η(X) = 2(Dϕ, Xj(ϕ)) − 2η(X). We consider the space of all possible types of structures T ∗M 6 ⊗ ϕ⊥ ∋ ∇ϕ, where ϕ⊥ = Rj(ϕ) ⊕ {X ·ϕ | X ∈ TM 6} is the orthogonal complement of ϕ in Σ. The restricted Clifford multiplication m is defined by m : T ∗M 6 ⊗ ϕ⊥ → Σ. Let π : Spin(6) → SO(6) be the usual projection. For h ∈ Spin(6) we have
m(π(h)η ⊗ hϕ∗) = hηh−1hϕ∗ = hm(η ⊗ ϕ∗) and m is Spin(6) equivariant and thus SU(3) equivariant. Comparing the dimensions of the modules given in (I.4) and the ones of Lemma 3.6 we see that χ223¯ ⊂ Ker(m) and with
Dϕ = 6λj(ϕ) for S = λJϕ and Dϕ = −6µϕ for S = µId we have the correspondences χ1 → Rj(ϕ) and χ1¯ → Rϕ and get (Dϕ, j(ϕ)) = 6λ and (Dϕ, ϕ) = −6µ. For a closer look at χ45 we recall that {Jϕeiϕ, ϕ, j(ϕ)} is a basis of Σ for some local orthonormal frame ei, hence
∑6 Dϕ = (Dϕ, Jϕeiϕ)Jϕeiϕ + (Dϕ, ϕ)ϕ + (Dϕ, j(ϕ))j(ϕ). i=1 With Lemma 3.8 we conclude
∑6 1 1 Dϕ = [ δω(e ) + η(e )]e j(ϕ) + 6λj(ϕ) − 6µϕ = ( δω + η)j(ϕ) + 6λj(ϕ) − 6µϕ. 2 i i i 2 i=1
Thus as the image of the Clifford multiplication, the R6 component of Σ− is determined by δω + 2η. 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 15
Theorem 3.9. On a 6-dimensional Riemannian spin manifold there exists a spinor ϕ ∈ Σ of length one in the kernel of the Dirac operator
Dϕ = 0 if and only if it admits an SU(3) structure of type χ22345¯ with the restriction δω = −2η on the χ4 and the χ5 part of the intrinsic torsion. The χ1- and the χ1¯-component of the intrinsic torsion are given by (Dϕ, j(ϕ)) = 6λ and (Dϕ, ϕ) = −6µ.
Remark 3.10. One could also look at the Spin(6) invariant Twistor operator P restricted to Σ+ ⊃ iRϕ ⊕ ϕ⊥, the projection on the kernel of the nonrestricted Dirac operator. But since the (nonrestricted) Dirac operator mixes the χ4 part with the TM ⊗ ϕ part, this projection is not useful to us.
As one can see in Lemma 3.6, the χ11¯ part are also determined by the trace of JϕS and S. Indeed we have tr(S) = −(Dϕ, ϕ) = 6µ and tr(JϕS) = −(Dϕ, j(ϕ)) = −6λ.
The linear combination χ4 +2χ5 vanishing in the theorem also shows up (up to volume choice) in work of Cardoso et al. [CCD03] and plays a role in supersymmetric compactifications of heterotic string theory.
Example 3.11. Consider the Lie algebra g = span{e1, . . . , e6} with structure equations dei = 0 if i = 3, 4, 5 and de1 = e3 ∧ e4 + 2e3 ∧ e5, de2 = e4 ∧ e5, de6 = e5 ∧ e1 + e2 ∧ e3.
Since the structure constants are rational the corresponding 1-connected Lie group G has a co-compact lattice Γ. We consider the spin structure on M 6 = G/Γ given by the pointwise construction of the Clifford algebra of the global vector fields ei given above and the SU(3) t ∇ − − structure determined by choosing ϕ = (1, 0, 0, 0, 0, 0, 0, 1) . With g( ei ej, ek) = dek(ei, ej) j i de (ek, ei) + de (ej, ek) this gives us 0 0 −2 0 1 0 1 −1 0 S = 1 0 0 −1 , η = e1 2 −1 0 0 −1 0 1 1 0 and it is not hard to see that Dϕ = 0. The structure has type χ22345¯ , and the presence of the components 4 and 5 is reflected in the non-vanishing η. Example 3.12. We look at further example 3 of section 4 of [CS02]: The nilpotent 3-step Lie algebra given by
dei = 0, if i = 1, 2, 4, 5; de3 = e2 ∧ e5 and de6 = e1 ∧ e4 − e2 ∧ e3.
Again we calculate ∇ and get ∇ − ∇ ∇ − ∇ ∇ ∇ − e1 = E46, e2 = E35 + E36, e3 = E25 E26, e4 = E16, e5 = E23, e6 = E14 E23 − 1 where Eijei = ej and Eijej = ei. With the lift 2 eiej of Eij one gets 0 −1 −1 −1 1 1 1 S = 1 0 2 −1 0 and η = 0. This is a typical example of a half flat SU(3) structure. 16 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
Notation 3.13. We decompose the intrinsic endomorphism into
S = λJϕ + µId + S2 + S34 where JϕS2 = S2Jϕ, JϕS34 = −S34Jϕ and the trace of S2 and JϕS2 vanishes.
We shall now prove that the Niejenhuis tensor determines the χ112¯ 2¯-component in the following way. Lemma 3.14. − J ◦ ◦ − J ◦ ◦ N(X,Y,Z) = 2[ψϕ ((Jϕ S + S Jϕ)X,Y,Z) ψϕ ((Jϕ S + S Jϕ)Y,X,Z)].
Thus if the structure is of type χ11345¯ , the Niejenhuis tensor is given by J − N(X,Y,Z) = 8[λψϕ (X,Y,Z) µψϕ(X,Y,Z)].
Proof. We have g((∇X Jϕ)Y,Z) = −(∇X ω)(Y,Z) and get with Lemma 3.3 − ∇ ∇ − ∇ ∇ N(X,Y,Z) = ( X ω)(JϕY,Z) + ( Y ω)(JϕX,Z) ( JϕX ω)(Y,Z) + ( JϕY ω)(X,Z) − J J − J J = 2[ ψϕ (SX,JϕY,Z) + ψϕ (SY,JϕX,Z) ψϕ (SJϕX,Y,Z) + ψϕ (SJϕY,X,Z)] − J J − J J = 2[ ψϕ (JϕSX,Y,Z) + ψϕ (JϕSY,X,Z) ψϕ (SJϕX,Y,Z) + ψϕ (SJϕY,X,Z)] − J J = 2[ ψϕ ((JϕS + SJϕ)X,Y,Z) + ψϕ ((JϕS + SJϕ)Y,X,Z)].
Furthermore for S = λJϕ + µId + S34 we have
JϕS + SJϕ = Jϕ(S34 + λJϕ + µId) + (S34 + λJϕ + µId)Jϕ = −2λId + 2µJϕ. and get − J − J − N(X,Y,Z) = 2[ ψϕ (( 2λId + 2µJϕ)X,Y,Z) + ψϕ (( 2λId + 2µJϕ)Y,X,Z)] J − J − = 4[ψϕ ((λId µJϕ)X,Y,Z) + ψϕ (( λId + µJϕ)Y,X,Z)] J − J − J J = 4[λψϕ (X,Y,Z) µψϕ (JϕX,Y,Z) λψϕ (Y,X,Z) + µψϕ (JϕY,X,Z)] J − = 8[λψϕ (X,Y,Z) µψϕ(X,Y,Z)]. This proves the claim.
The χ1134¯ part of the intrinsic torsion is given by dω in the following way: Lemma 3.15. With Notation 3.13 we have XYZ J J dω(X,Y,Z) = 6λψϕ(X,Y,Z) + 6µψϕ (X,Y,Z) + 2 S ψϕ (S34(X),Y,Z).
XYZ XYZ ∇ J Proof. We have dω(X,Y,Z) = S ( X ω)(Y,Z), The fact that S ψϕ (S2(X),Y,Z) vanishes corresponds to dω = 0 in the class χ225¯ . To get additional equations in terms of the corresponding spinor ϕ, we need the following lemma. Lemma 3.16. The intrinsic torsion (S, η) of a Riemannian spin manifold (M 6, g, ϕ) satisfies the following properties:
⇐⇒ ∇ − ∇ SJϕ = JϕS (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), − ⇐⇒ ∇ ∇ SJϕ = JϕS (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ),
S is symmetric ⇐⇒ (X∇Y ϕ, ϕ) = (Y ∇X ϕ, ϕ),
S is skew-symmetric ⇐⇒ (X∇Y ϕ, ϕ) = −(Y ∇X ϕ, ϕ). 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 17
Proof. With (JϕS(X)ϕ, Y ϕ) = (SJϕ(X)ϕ, Y ϕ) if and only if ∇ − ∇ (JϕY X ϕ, ϕ) = (Y Jϕ(X)ϕ, ϕ), we prove the first two equivalences. Since ϕ, j(ϕ) ⊥ Y ϕ for any Y ∈ TM 6, we obtain
g(S(X),Y ) = (∇X ϕ, Y ϕ) and g(X,S(Y )) = (∇Y ϕ, Xϕ).
Thus we conclude the formulas for symmetric and skew symmetric S. Theorem 3.17. The classification of SU(3) structures in terms of the defining spinor ϕ is given in the following table, where η is the one-form given by η(X) := (∇X ϕ, j(ϕ)) and the functions 1 − 1 λ and µ are given by 6 (Dϕ, j(ϕ)) and 6 (Dϕ, ϕ).
class spinorial equation
χ1 ∇X ϕ = λXj(ϕ) for λ ∈ R
χ1¯ ∇X ϕ = µXϕ for µ ∈ R ∇ − ∇ ∇ ∇ χ2 (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), (Y X ϕ, j(ϕ)) = (X Y ϕ, j(ϕ)), λ = η = 0 ∇ ∇ ∇ − ∇ χ2¯ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), (Y X ϕ, j(ϕ)) = (X Y ϕ, j(ϕ)), µ = η = 0 ∇ ∇ ∇ ∇ χ3 (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), (Y X ϕ, j(ϕ)) = (X Y ϕ, j(ϕ)), and η = 0 ∇ − ∇ ∇ − ∇ χ4 (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), (Y X ϕ, j(ϕ)) = (X Y ϕ, j(ϕ)) and η = 0
χ5 ∇X ϕ = (∇X ϕ, j(ϕ))j(ϕ)
χ11¯ ∇X ϕ = λXj(ϕ) + µXϕ ∇ − ∇ χ22¯ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ), λ = µ = 0 and η = 0 ∇ − ∇ χ225¯ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ) and λ = µ = 0 ∇ − ∇ χ112¯ 2¯ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ) and η = 0 ∇ − ∇ χ112¯ 25¯ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ)
χ223¯ Dϕ = 0 and η = 0
χ112¯ 23¯ (Dϕ, Xϕ) = 0 and η = 0
χ112¯ 234¯ (∇X ϕ, j(ϕ)) = 0
χ2235¯ (Dϕ, Xj(ϕ)) = η(X) and λ = µ = 0
χ112¯ 235¯ (Dϕ, Xj(ϕ)) = η(X) ∇ ∇ χ34 (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ) and η = 0 ∇ ∇ χ345 (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ)
χ22345¯ λ = µ = 0
χ1¯23¯ (X∇Y ϕ, ϕ) = (Y ∇X ϕ, ϕ) and η = 0
Proof. We first prove that λ and µ in χ1 and χ1¯ are constant. In χ1 we have S = λJϕ and thus ∇X (ϕ+j(ϕ)) = −λX(ϕ+j(ϕ)). Since a nearly K¨ahlerstructure (type χ115¯ ) is given by a Killing spinor [Gr90], this λ must be constant. In the case χ1¯ the spinors ϕ and j(ϕ) themselves are 18 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
Killing spinors with Killing constants µ and −µ.
We combine the results of Lemma 3.16 as follows. By Lemma 3.6, a structure is of type χ2 if S is skew symmetric, commutes with Jϕ, JϕS = SJϕ and the trace of JϕS and η vanish. With Lemma 3.16 we get the condition (X∇Y ϕ, ϕ) = −(Y ∇X ϕ, ϕ) under which the equation ∇ − ∇ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ) is equivalent to ∇ − ∇ ∇ − ∇ ∇ (Y X ϕ, j(ϕ)) = (JϕY X ϕ, ϕ) = (X JϕY ϕ, ϕ) = (JϕX Y ϕ, ϕ) = (X Y ϕ, j(ϕ)). The other classes are to be calculated similarly, making extensive use of Lemmas 3.8, 3.16. It makes little sense to compute all possible combinations (in principle, 27), so we have listed here only some that raise interest. Others can be inferred by arguments of the following sort: suppose we want to show that class χ124 has (X∇Y ϕ, ϕ) = −(Y ∇X ϕ, ϕ) and η = 0 as defining equations. From Lemma 3.6 we know χ124 is governed by the fact that S is skew symmetric, and at the same time η controls χ5, whence the claim. Other identities are given by the following kind of argument. Assume we want to show that the equation
XYZ XYZ ∇ J S (YZ X ϕ, j(ϕ))+ S η(X)g(Y,Z) = 3λψϕ(X,Y,Z) + 3µψϕ (X,Y,Z)
J defines the class χ112¯ 25¯ . From Lemma 3.15 we know that dω = 6λψϕ(X,Y,Z) + 6µψϕ (X,Y,Z) XYZ defines this class, so we finish with dω(X,Y,Z) = S (∇X ω)(Y,Z) and the first equality in the proof of Lemma 3.8. J Additionally to dω one can compute dψϕ and dψϕ in terms of ϕ and then use the correspondences in [CS02] to get more equations (e.g. a structure is half flat, of type χ1¯23¯ , if dω ∧ ω = 0 and J dψϕ = 0). Remark 3.18. (i) The proof shows that the real Killing spinors of an SU(3)-structure of type χ11¯ (with Killing constants ±|λ|) have the form
ϕ ± j(ϕ) in the case χ1 and ϕ and j(ϕ) in the case χ1¯
Now, in class χ123¯ we have the constraint Dϕ = fϕ, so ϕ is an eigenspinor with eigenfunction f. (One can change ϕ such that one achieves an eigenspinor even in χ112¯ 23¯ .) But we are not aware of a nice argument, showing that f is constant as in the nearly K¨ahlercase. 2 2 (ii) Rescaling ϕ to f1ϕ+f2j(ϕ) by functions f1, f2 with f1 +f2 = 1 affects the structure as follows: 2 2 df2 the intrinsic tensors transform as S (f − f )S + 2f1f2Jϕ ◦ S and η η + . (cf. Section 1.1 1 2 f1 6 ± for the case where f = h is constant on M ). The χ5-component varies, and χi , i = 1, 2 change, too, cf. [CS02]. Therefore, if we are looking at a Killing spinor (corresponding to SU(3)-type df χ ¯ ), then f necessarily determines the fifth component η = − 2 . 115 f1 (iii) It is fairly evident (cf. [CS02]) that the effect of a rotation S 7→ JS is the exchange + ←→ − χj χj , j = 1, 2, while the other ones are untouched. Example 3.19. Schoemann describes in [Sc06] the almost complex structures on the twistor space CP3 over the manifold S4. He uses the construction of [BFGK91, Section 3.3] to consider 3 a family of metrics gt on CP given by
∗ gt := π gˆ + tg,˜ where π∗gˆ is the metric on S4 beeing pulled back by π : CP3 → S4 and (˜g, J˜) is the standard K¨ahlerstructure on the fibre S2 over a point in S4. One defines an almost complex structures J 3 over a point Jx in CP by −1 J = π ◦ Jx ◦ π − J.˜ 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 19
Schoemann describes the types of the structure in dependence of t. The type of the corresponding 1 SU(3) structure is χ112¯ 25¯ for general t and χ11¯ for t = 2 . As the homogenous space SO(5)/U(2), this structure is given by a invariant spinor ϕ. In particular the general equation for such a structure is ∇ − ∇ (JϕY X ϕ, ϕ) = (Y JϕX ϕ, ϕ) 1 for general t and the Killing equation for t = 2 . We continue the calculation of [BFGK91] and get √ √ √ √ t t t t 1 − t 1 − t ∇X ϕ = S(X)ϕ where S = diag( , , , , √ , √ ) 2 2 2 2 2 t 2 t in particular η = 0. There is another structure J + on CP3, which is K¨ahlerat the value t = 1. Unfortunately there is no invariant spinor in our description of CP3 as SO(5)/U(2) describing J +. 3 In the same way, one could look at the Flag manifold F1,2 = U(3)/U(1) as the twistor space of CP2. In this case we know from [AGI98] that all the complex structure are invariant.
3.4 Adapted connections Let (M 6, g, ϕ) be a 6 dimensional SU(3) manifold with Levi-Civita connection ∇. As we are interested in non integrable structures, ∇ϕ ≠ 0, we are looking for a metric connection that preserves the SU(3) structure. In Section 3.2 we introduced the Levi-Civita connection Z = Z˜ + Γ, where Γ is the intrinsic torsion. We will consider the SU(3)-connection Z˜. This connection is often called canonical connection. The space of all metric connections ∇¯ is isomorphic to the space of all (2, 1) tensors Ag := A ∈ 6 2 6 TM ⊗ Λ (TM ) by ∇˜ X Y = ∇X Y + A(X,Y ). Let S be the intrinsic endomorphism and η the intrinsic 1-form of the SU(3) structure on M 6. We define the map
2 Ξ: TM 6 ⊕ End(TM 6) → Ag, (η, S) 7→ −Syψ + η ⊗ ω. ϕ 3 ∇n ∇ 6 Then X Y := X Y + Ξ(η, S) defines a metric connection on M and with Lemma 3.2 we get 1 1 ∇n ϕ = ∇ ϕ − ψ (S(X), ., .) · ϕ + η(X)ω · ϕ X X ϕ 2 3 = S(X) · ϕ + η(X)j(ϕ) − S(X) · ϕ − η(X)j(ϕ) = 0.
Thus ∇n is an SU(3) connection. The space Ag splits under the representation of SO(6) in (see page 51 of [Ca25] and [AF04])
Ag = TM 6 ⊕ Λ3(TM 6) ⊕ T .
The three classes are given in the following table
class name relation
T M vectorial torsion ∃V ∈ TM s.t. A(X,Y,Z) = g(X,Y )g(Z,V ) − g(X,Z)g(Y,V ) Λ3(T M) (totally) skew symmetric torsion A(X,Y,Z) = −A(Y,X,Z) XYZ ∑ T cyclic traceless torsion S A(X,Y,Z) = 0 and i A(ei, ei,X) = 0 20 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
for some local basis e1 with i = 1..7 and X,Y,Z ∈ TM. A metric connection is said to be of a certain type, if its torsion is of this type. With an appropriate computer algebra system one calculates the map Ξ at one point and gets
Lemma 3.20. On a 6-dimensional SU(3) manifold the type of the connection ∇n is given in the following table.
6 type of M χ11¯ χ22¯ χ3 χ4 χ5 type of ∇n Λ3(R6) T Λ3(R6) ⊕ T R6 ⊕ Λ3(R6) ⊕ T R6 ⊕ Λ3(R6) ⊕ T
The projection to the skew symmetric part of the torsion given in the previous lemma gives the characteristic connection of the SU(3) structure. We are interested in finding out whether and when an SU(3) structure (M 6, g, ϕ) admits a char- acteristic connection. Any such must be the (unique) characteristic connection of the underlying c U(3)-structure with the additional condition ∇ ψϕ = 0 and thus we know from [FI02] that the χ22¯-part of the intrinsic torsion of this corresponding almost hermitian structure vanishes. So the SU(3)-type must necessarily be χ11345¯ . Additionally we have Lemma 3.21. Given an SU(3) manifold (M 6, g, ϕ), a connection with skew torsion ∇˜ is char- acteristic if and only if it preserves the spinor ϕ,
∇˜ = ∇c ⇐⇒ ∇˜ ϕ = 0.
Proof. Obvious, but just for the record: ∇c is an SU(3)-connection, and SU(3) = Stab(ϕ) forces ϕ to be parallel. Conversely, if ϕ is ∇˜ -parallel, the connection must preserve any tensor defined in terms of the spinor, like ω and ψϕ, cf. Lemma 3.3. To conclude one must recall that the characteristic connection is unique, see [AFH13].
To obtain the ultimative sufficient and necessary condition we need to impose an additional constraint on the torsion components χ4, χ5: Theorem 3.22. A Riemannian spin manifold (M 6, g, ϕ) admits a characteristic connection if 1 −∗ ∗ and only if it has type χ11345¯ and η = 4 δω, where δ = d is the co-derivative. Proof. Let ∇c be the U(3)-characteristic connection, T its torsion. We shall determine in which cases ∇cϕ = ∇cj(ϕ) = 0. First of all
∇c − ∇c − ∇c 0 = ( X ω)(Y,Z) = 2( X ϕ, ZY j(ϕ)) 2g(Y,Z)( X ϕ, j(ϕ)). ∇c ⊥ ⊥ Thus we have ( X ϕ, ZY j(ϕ)) = 0 if Y Z. But for all Y Z the spinors Y Zj(ϕ) span the space orthogonal to ϕ and j(ϕ). In conclusion, ∇c is characteristic for the SU(3)-structure if and ∇c − only if ( X ϕ, j(ϕ)) = 0. Now choose a local adapted basis e1, .., e6 with Jϕei = ei+1, i = 1, 3, 5. Using the formula 1 ∇c ϕ = ∇ ϕ + (Xy T )ϕ X X 4 and ω(X,Y ) =∑−(XY ϕ, j(ϕ)) we arrive at 4η(X) = −(Xy T ϕ, j(ϕ)) = ω(Xy T ) = −T (X, ω) = − − T (ω, X) = i=1,3,5 T (ei,Jϕei,X). From T (ei, Jei,X) = T (Jei, ei,X) = T (ei+1, Jei+1,X) for i = 1, 3, 5, we infer ∑ ∑ − 1 6 − 6 ∇ 4η(X) = 2 i=1 T (ei, Jei,X) = i=1( ei ω)(ei,X) = δω(X) ∇c ∇ − 1 because 0 = ( X ω)(Y,Z) = ( X ω)(Y,Z) 2 (T (X,JϕY,Z) + T (X,Y,JϕZ)). Corollary 3.23. If an SU(3) structure is of type 3. SPECIAL ALMOST HERMITIAN STRUCTURES IN DIMENSION 6 21
• χ1 ⊕ χ3 there is a unique characteristic connection,
• χ2 there is no characteristic connection,
• χ4 ⊕ χ5 there is a unique characteristic connection if and only if δω = 4η. The next theorem will give an explicit formula for the torsion of ∇c. It relies on the computation for the Nijenhuis tensor of Lemma 3.14. 6 Suppose M has type χ11345¯ , and decompose the intrinsic endomorphism into
S = λJϕ + µId + S34, as explained in Notation 3.13.
6 Theorem 3.24. Suppose (M , g, ϕ) has type χ11345¯ . Then the characteristic torsion of the induced U(3) structure reads
XYZ J − − T (X,Y,Z) = 2λψϕ (X,Y,Z) 2µψϕ(X,Y,Z) 2 S ψϕ(S34(X),Y,Z).
1 If η = 4 δω then T is the characteristic torsion of the SU(3) structure as well. Proof. With Lemma 3.15 we have
XYZ ◦ J − dω Jϕ(X,Y,Z) = 6λψϕ (X,Y,Z) 6µψϕ(X,Y,Z) + 2 S ψϕ(S34(X),Y,Z) The formula T = N − dω ◦ J (see [FI02]) together with Lemma 3.14 gives the result. Remark 3.25. Another way of proving Theorem 3.24 is given by the explicit formula of the intrinsic∑ torsion in Proposition 3.4. For an arbitrary basis e1, .., e6 the torsion T is then given − y ⊗ → ⊥ by i prsu(3)⊥ (ei T ) ei = 2Γ, where prsu(3)⊥ denotes the projection so(6) su(3) with ⊥ so(6) = su(3) ⊕ su(3) . This method is used for G2 structures in Theorem 4.14. One considers the map
TM ∗ ⊗ m κ / Λ3(T M ∗) Θ / T M ∗ ⊗ m
XYZ − 2 / 1 2 ψ (SX,Y,Z) + 3 η(X)ω(Y,Z) 3 S (ω(SX,Y,Z) + 3 η(X)ω(Y,Z)) ∑ / ⊗ y T i ei (ei T )m.
◦ 1 With a computer algebra system one shows for example Θ κ|χ3 = 3 Idχ3 and obviously we have ◦ Θ κ|χ1 = Idχ1 . To conclude Theorem 3.24 one would also need to calculate the mixing 4 and the 5 parts, which thus is a little harder then the corresponding calculation in the G2 case as in section 4.4.
Manifolds with parallel torsion are of particular interest. A motivation came from [Fr98]. By a Theorem of Kirichenko, nearly K¨ahlermanifolds are of this type. Parallel torsion is used in [AF08] in dimensions 4 and 5, in [Fr07] in dimension 7 and in [Pu09] in dimension 8. Suppose M 6 admits a characteristic connection ∇c. If S is ∇c-parallel, then T c is parallel with respect to ∇c. We now consider the torsion to be ∇c-parallel. Since ∇cϕ = 0 with [ABBK13] or [FI02] an easy computation shows ∑ 1 y ∧ y c Corollary 3.26. Let σT be the 4-form given by 2 i(ei T ) (ei T ) and let Ric be the Ricci curvature with respect to the connection ∇c. Then 1 (XyRicc) · ϕ = (XydT ) · ϕ = (Xyσ ) · ϕ. 2 T 22 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
In particular in the case χ11¯ we have J − 2 2 ∗ σT = λd(ψϕ ) µdψϕ = 12(λ + µ ) ω. Example 3.27. We take the real 6-manifold M 6 = SL(2, C) and view it as the reductive space SL(2, C) × SU(2) = G/H SU(2) with diagonal embedding. Let g, h be the Lie algebras of G and H, and set g = h ⊕ m, so that m = {(A, B) ∈ g | A − A¯t = 0, tr(A) = 0,B + B¯t = 0, tr(B) = 0}. The almost complex structure J(A, B) = (iA, iB) ∇c ∇ 1 defines a U(3) structure of type χ3 (see [AFS05]).The characteristic connection = + 2 T preserves a spinor ϕ, so ∇c is also characteristic for the induced SU(3) structure, which is of type χ35. But by Theorem 3.22 we have η = 0, so actually the SU(3) type is χ3. But then ϕ is harmonic. More general it is easy to see that Corollary 3.28. For any spinor ϕ the condition ϕ ∈ Ker D is equivalent to T ϕ = 0 whenever ∇c exists. Since for our example T ϕ = 0 was shown in [AFS05] the equation Dϕ = 0 may be employed if more convenient. From Theorems 3.9, 3.22 we also know, that if ∇c exists and Dϕ = 0 the SU(3) structure must be of type χ3. This example shows, that there are SU(3) structures of type different to χ115¯ (in this case χ3) with ∇cT = 0.
4 G2 geometry
Let (M, g, ) be a 7-dimensional oriented Riemannian manifold. M is said to carry a G2 structure if it admits a reduction to G2 ⊂ SO(7); alternatively, this amounts to the choice of a generic 3-form Ψ. With respect to a local orthonormal frame e1, . . . , e7, such a 3-form can locally be written as Ψ = e123 + e145 + e167 + e246 − e147 − e347 − e356. Here and subsequently, we do not distinguish between vectors and covectors and denote the ∧ ∧ k-form ei1 .. eik by ei1..ik . G2 manifolds were classified by Fern´andezand Gray in [FG82] into four classes W1 ⊕ W2 ⊕ W3 ⊕ W4. Friedrich and Ivanov proved that there is a characteristic connection if and only if the structure is of class W1 ⊕ W3 ⊕ W4; these manifolds are sometimes called G2 manifolds with torsion or G2T manifolds for short. In [FI02] a concrete description of the torsion can be found (we do not need the explicit formula here). We will often use the skew symmetric endomorphism P (X,.) introduced in [FG82], Ψ(X,Y,Z) = g(X,P (Y,Z)).
We will use this description mostly in Section 4 of Chapter II, since there we need a description of a G2 structure without spinors. Since here we lift a G2 structure to a Spin(7) structure in dimension 8, additionally the (2, 1)-tensor P we have P¯, which is the (3, 1)-tensor, defining the Spin(7) structure. This makes the notation for the G2 tensor P more convenient than the often used (for example in this thesis all through this section and Section 1 of Chapter II) notation P (X,Y ) = X × Y. 4. G2 GEOMETRY 23
In Section 1 of Chapter II we use the description via spinors, which again gives a new viewpoint on G2 structures: As in Section 3 we have G2 as a simply connected subgroup of SO(7) and are thus able to construct the lift G2 ⊂ Spin(7), where again G2 is the stabilizer of a spinor. To better understand this we take a closer look at the linear algebra in dimension 7.
4.1 Linear algebra in dimension 7
7 We consider R and the corresponding Spin(7) representation ∆7 = ∆6. As in the 6-dimensional case, we have a real structure α. Since α commutes with the Clifford multiplication, we can fix the following real 8-dimensional Spin(7) representation
∆ := {ϕ ∈ ∆7 | α(ϕ) = ϕ}.
We denote the corresponding real scalar product by ( .,. ). For any spinor ϕ ∈ ∆ and X ∈ R7 we have (X · ϕ, ϕ) = −(ϕ, X · ϕ) = −(X · ϕ, ϕ) and thus X · ϕ ⊥ ϕ for all X ∈ R7. The space {X · ϕ | X ∈ R7} is 7-dimensional and thus {X · ϕ | X ∈ R7} = ϕ⊥. Given a spinor ϕ of length one, we can define a 3-form
Ψϕ(X,Y,Z) = (X · Y · Z · ϕ, ϕ) (I.6)
7 7 for X,Y,Z ∈ R . This 3-form is generic and normalized: For a orthonormal basis e1, .., e7 of R for any i, j = 1, .., 7 with i ≠ j we have ∑ Ψϕ(ei, ej, ek) = 1. k
Such a 3-form has stabilizer G2 in SO(7) and since G2 is simple it can be lifted to Spin(7) and is also the stabilizer of the spinor ϕ. Vice versa given a generic, normalized 3-form Ψ we can define a spinor ϕ of length one via equation (I.6). Note that this spinor is only defined up to a multiplication of ±1. We get
Lemma 4.1. There is a one to one correspondence of
• generic normalized 3-forms in R7
• reductions of SO(7) to G2
• reductions of Spin(7) to G2
• one dimensional subspaces in ∆.
7 Thus the space of G2 structures on R is given by
RP(∆) = RP(7) = SO(7)/G2.
Useful for further calculations is the following
Lemma 4.2. We have
∗ ∗ ∗ Ψϕ · ϕ = 7ϕ, Ψϕ · ϕ = −ϕ if ϕ ⊥ ϕ , (X Ψϕ) · ϕ = −3X · ϕ. 24 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
4.2 G2 manifolds Let (M 7, g, ϕ) always be a seven dimensional Riemannian manifold with Levi-Civita connection ∇ equipped with a global spinor ϕ of length one, inducing a G2-structure given by a 3-form Ψϕ. Due to the fact that {Xϕ | X ∈ TM 7} = ϕ⊥ there exists an endomorphism S : TM 7 → TM 7 satisfying ∇X ϕ = S(X)ϕ.
We consider the corresponding 3-form Ψϕ(X,Y,Z) := (X · Y · Z · ϕ, ϕ) and the cross product
g(X × Y,Z) = Ψϕ(X,Y,Z). Lemma 4.3. The cross product satisfies (X × Y )ϕ = −XY ϕ − g(X,Y )ϕ Proof. We have
(Y Zϕ, Xϕ) = −(XY Zϕ, ϕ) = −Ψϕ(X,Y,Z) = −g(X,Y × Z) = −((Y × Z)ϕ, Xϕ). Thus we have ((X × Y )ϕ, ϕ∗) = (−XY ϕ − g(X,Y )ϕ, ϕ∗) for all ϕ∗ ⊥ ϕ. For ϕ∗ = ϕ we have ((X × Y )ϕ, ϕ∗) = 0 and the equality is satisfied since g(XY ϕ, ϕ) = −g(X,Y )(ϕ, ϕ). We cite (see also [FG82] and the beginning of Section 4 in Chapter II) Lemma 4.4.
∗Ψϕ(V, W, X, Y ) = Ψϕ(V, W, X × Y ) − g(V,X)g(W, Y ) + g(V,Y )g(W, X). and conclude Lemma 4.5. The map S is given by
(∇V Ψϕ)(X,Y,Z) = 2 ∗ Ψϕ(S(V ),X,Y,Z). Proof. We calculate
(∇V Ψϕ)(X,Y,Z) = (XZY ∇V ϕ, ϕ) + (XY Zϕ, ∇V ϕ) = (XZYS(V )ϕ, ϕ) + (XY Zϕ, S(V )ϕ) = (XZYS(V )ϕ, ϕ) − (S(V )XY Zϕ, ϕ) = (XZYS(V )ϕ, ϕ) + (XS(V )Y Zϕ, ϕ) + 2g(S(V ),X)(Y Zϕ, ϕ) = (XZYS(V )ϕ, ϕ) − (XYS(V )Zϕ, ϕ) − 2g(S(V ),Y )(XZϕ, ϕ) +2g(S(V ),X)(Y Zϕ, ϕ) = (XZYS(V )ϕ, ϕ) + (XYZS(V )ϕ, ϕ) + 2g(S(V ),Z)(XY ϕ, ϕ) −2g(S(V ),Y )(XZϕ, ϕ) + 2g(S(V ),X)(Y Zϕ, ϕ) = 2(XZYS(V )ϕ, ϕ) − 2g(S(V ),Z)g(X,Y ) + 2g(S(V ),Y )g(X,Z) −2g(S(V ),X)g(Y,Z). With Lemma 4.4 we get
2 ∗ Ψϕ(S(V ),X,Y,Z) = −2 ∗ Ψϕ(X,Y,Z,S(V )) = −2[(XY (Z × S(V ))ϕ, ϕ) − g(X,Z)g(S(V ),Y ) +g(X,S(V ))g(Y,Z)] = 2(XYZS(V )ϕ, ϕ) + 2g(Z,S(V ))(XY ϕ, ϕ) +2g(X,Z)g(S(V ),Y ) − 2g(X,S(V ))g(Y,Z) = 2(XYZS(V )ϕ, ϕ) − 2g(Z,S(V ))g(X,Y ) +2g(X,Z)g(S(V ),Y ) − 2g(X,S(V ))g(Y,Z). 4. G2 GEOMETRY 25
Proposition 4.6. The intrinsic torsion of the G2-structure Ψϕ(X,Y,Z) = (XY Zϕ, ϕ) is
2 Γ = − Sy Ψ 3 ϕ where Sy Ψϕ(X,Y,Z) = Ψϕ(S(X),Y,Z).
∇ 1 Proof. As in the proof of Theorem 3.4 we have X ϕ = 2 Γ(X)ϕ and
1 ∇ ϕ = S(X)ϕ = − (S(X)yω)ϕ. X 3
∈ y ⊥ ⊕ ⊥ For any X TM, the 2-form X ω is in g2 with respect to the splitting so(7) = g2 g2 . Thus 7→ − 1 y ⊥ X 3 (S(X) )ω is a g2 valued 1-form. Again, the same argument as in the proof of Theorem − 2 y 3.4 shows that this implies Γ(X) = 3 S(X) ω.
∗ ⊥ To classify G2 structures we look at a decomposition of ∇ϕ ∈ TM ⊗(ϕ ) due to the identification
∗ ∼ ∗ ⊥ TM ⊗ TM = TM ⊗ (ϕ ), η ⊗ X 7→ η ⊗ X · ϕ.
Then the splitting of TM ∗ ⊗ (ϕ⊥) is given by ∑ ∗ ⊥ TM ⊗ (ϕ ) = { ei ⊗ ei · ϕ} ∑i ⊕ { aijei ⊗ ej · ϕ | (aij) ∈ g2} ∑ij ⊕ { aijei ⊗ ej · ϕ | (aij) is traceless symmetric } ∑ij 7 ⊕ { aijei ⊗ ej · ϕ | (aij) ∈ R }. ij and corresponds to the decomposition of endomorphisms of R7
7 7 7 End(R ) = R · Id ⊕ S0(R ) ⊕ g2 ⊕ R ,
7 7 where S0(R ) denotes the symmetric, traceless endomorphisms of R . Identifying the dimensions one can compare the classes to the classes given in [FG82].
Lemma 4.7. We have the following correspondence of classes of G2 structures
class description dimension
W1 S = λ · Id 1
W2 S ∈ g2 14
7 W3 S ∈ S0(R ) 27
7 7 W4 S ∈ R = {V yΨϕ | V ∈ R } 7
In particular, S is symmetric if and only if S ∈ W1 ⊕ W3. 26 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
4.3 Spin formulation
∗ ∼ ∗ ⊥ Since T M 7 ⊗ TM 7 = T M 7 ⊗ ϕ , η ⊗ X 7→ η ⊗ Xϕ is an isomorphism we can read the known ∗ 7 ⊥ decomposition T M ⊗ ϕ = W1 ⊕ W2 ⊕ W3 ⊕ W4 via ϕ.
Notation 4.8. As in the SU(3) case we will shorten for example W1 ⊕ W2 ⊕ W4 to W124.
∗ 7 ⊥ The restricted Clifford product m : T M ⊗ ϕ → ∆ splits the space W1234 as follows.
7 Theorem 4.9. Let (M , g, ϕ) be a Riemannian G2 manifold. Then ϕ is a harmonic spinor
Dϕ = 0
if and only if the underlying G2-structure has type W23.
⊥ Proof. First, the spin representation splits as ∆ = Rϕ ⊗ ϕ = W1 ⊕ W4, so we may write the intrinsic-torsion space as
7 ⊥ TM ⊗ ϕ = ∆ ⊕ W23.
∗ 7 ⊥ ∗ 7 ⊥ Yet the restricted Clifford product m : T M ⊗ϕ → ∆ is G2-equivariant. Since T M ⊗(ϕ ) = W ⊕ W ⊕ W ⊕ W R ⊕ ⊕ 2 ∗ 7 ⊕ 7 W 1 2 3 4 = ϕ g2 S0 T M TM we have Ker m = 23, and the assertion follows from the definition D = m ◦ ∇.
Lemma 4.10. The W24 part of the intrinsic torsion in terms of ϕ is given by
1 ∇ − ∇ 2 δΨϕ(X,Y ) = (Xϕ, Y ϕ) (Y ϕ, X ϕ) + (Dϕ, XY ϕ) + g(X,Y )(Dϕ, ϕ).
Proof. To prove the claim we simply calculate, in some basis (ei)i=1..7, ∑ ∑ δΨ (X,Y ) = − (∇ Ψ )(e ,X,Y ) = − [(XY e ∇ ϕ, ϕ) + (XY e ϕ, ∇ ϕ)] ϕ ei ϕ i ∑ i ei i ei − − − ∇ ∇ = 2(XY Dϕ, ϕ) [ 2g(ei,Y )(Xϕ, ei ϕ) + 2g(ei,X)(Y ϕ, ei ϕ) ∇ +(eiXY ϕ, ei ϕ)]
= −2(XY Dϕ, ϕ) + 2(Xϕ, ∇Y ϕ) − 2(Y ϕ, ∇X ϕ) + (XY ϕ, Dϕ)
= 2(Dϕ, XY ϕ) + 2(Xϕ, ∇Y ϕ) − 2(Y ϕ, ∇X ϕ) + 2g(X,Y )(Dϕ, ϕ).
Theorem 4.11. The basic classes of G2-manifolds are described by the behaviour of the spinor ϕ as follows 4. G2 GEOMETRY 27
class spinorial equation
W1 ∇X ϕ = λXϕ with λ ∈ R
W2 ∇X×Y ϕ = Y ∇X ϕ − X∇Y ϕ + 2g(Y,S(X))ϕ
W3 (X∇Y ϕ, ϕ) = (Y ∇X ϕ, ϕ) and λ = 0
7 W4 ∃V ∈ T M : ∇X ϕ = XV ϕ + g(V,X)ϕ
W12 ∇X×Y ϕ = −14λ[Y ∇X ϕ − X∇Y ϕ + g(Y,S(X))ϕ − g(X,S(Y ))ϕ]
W13 (X∇Y ϕ, ϕ) = (Y ∇X ϕ, ϕ)
7 W14 ∃V,W ∈ TM : ∇X ϕ = XV W ϕ − (XV W ϕ, ϕ)
W23 Sϕ = 0 and λ = 0, or Dϕ = 0
W24 (X∇Y ϕ, ϕ) = −(Y ∇X ϕ, ϕ)
W34 3(Xϕ, ∇Y ϕ) − 3(Y ϕ, ∇X ϕ) = (Sϕ, XY ϕ) and λ = 0
W123 (Sϕ, Xϕ) = 0, or Dϕ = −7λϕ
W124 (Y ∇X ϕ, ϕ) + (X∇Y ϕ, ϕ) = −2λg(X,Y )
W134 3(Xϕ, ∇Y ϕ) − 3(Y ϕ, ∇X ϕ) = (S · ϕ, XY ϕ) − 7λg(X,Y )
W234 λ = 0
− 1 → R × where λ = 7 (Dϕ, ϕ): M is a real function and X Y denotes the cross product relative to Ψϕ.
Proof. For W1 there is actually nothing to prove, for the given equation is nothing but the Killing spinor equation characterising this type of manifolds [Fr80] (see also [BFGK91]). If S ∈ W2 then S is skew-symmetric, which implies S(X × Y ) = S(X) × Y + X × S(Y ). Then
∇X×Y ϕ = (−Y × S(X) + X × S(Y ))ϕ = YS(X)ϕ + g(Y,S(X))ϕ − XS(Y )ϕ − g(X,S(Y ))ϕ
= Y ∇X ϕ − X∇Y ϕ + 2g(S(X),Y )ϕ. By taking the dot product with ϕ we re-obtain that S is skew symmetric. For W3 we use Lemma 4.10: 1 ∇ − ∇ 2 δΨϕ(X,Y ) = (Dϕ, XY ϕ) + g(X,Y )(Dϕ, ϕ) + (Xϕ, Y ϕ) (Y ϕ, X ϕ). This fact together with Tr S = −(Dϕ, ϕ) allows to conclude. 7 7 Suppose S ∈ W4. The vector representation R is {V × . | V ∈ R }, so if S is represented by V we have ∇X ϕ = V × X = −V Xϕ − g(V,X)ϕ = XV ϕ + g(V,X)ϕ. As for the rest, we shall only prove what is not obvious. ∑ W 1 A structure is of type 23 if (Dϕ, ϕ) = 0 and 0 = 2 i,j δΨϕ(ei, ej)Ψϕ(ei, ej,X) (see [FG82]). This is equivalent to ∑ 0 = [(Dϕ, e e ϕ) − 7g(e , e )λ + (e ϕ, S(e )ϕ) − (e ϕ, S(e )ϕ)](e e Xϕ, ϕ) ∑i,j i j i j ∑i j j i i j = − (Dϕ, eiejϕ)(eiejϕ, Xϕ) + 2 (eiϕ, S(ej)ϕ)(eiejXϕ, ϕ). i,j i,j ∑ { | } ∗ Using the fact that eiejϕ i, j = 1..7 is a basis of ∆ we obtain i,j (ϕ , eiejϕ)eiejϕ = 6ϕ1 + ∗ (ϕ , ϕ)ϕ. Define ∑ Sϕ := g(ei,S(ej))eiejϕ i,j 28 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS and get 0 = −6(Dϕ, Xϕ) − 2(Sϕ, Xϕ). Thus 3Dϕ = −Sϕ holds on ϕ⊥. If λ = 0 we then have ∑ ∑ (Sϕ, ϕ) = g(ei,S(ej))(eiejϕ, ϕ) = − g(ei,S(ei)) = (Dϕ, ϕ) = 0.
A structure is of type W34 if (Dϕ, ϕ) = 0 and ∑ 1 × 3δΨϕ(X,Y ) = 2 i,j δΨϕ(ei, ej)Ψϕ(ei, ej,X Y ). Due to the calculation above the right-hand side equals
−6(Dϕ, (X × Y )ϕ) − 2(Sϕ, (X × Y )ϕ) = 6(Dϕ, XY ϕ) − 42g(X,Y )λ + 2(Sϕ, XY ϕ) + 2g(X,Y )(Sϕ, ϕ).
We have
3δΨϕ(X,Y ) = 6(Dϕ, XY ϕ) − 42g(X,Y )λ + 6(Xϕ, ∇Y ϕ) − 6(Y ϕ, ∇X ϕ) thus the defining equation is equivalent to
3(Xϕ, ∇Y ϕ) − 3(Y ϕ, ∇X ϕ) = (Sϕ, XY ϕ) − 7g(X,Y )λ and if λ = 0 we get 3(Xϕ, ∇Y ϕ) − 3(Y ϕ, ∇X ϕ) = (Sϕ, XY ϕ). As for W124, note that S satisfies
(Y ∇X ϕ, ϕ) + (X∇Y ϕ, ϕ) = −2g(X,Y )λ if it is skew. If symmetric, instead, it satisfies the equation if and only if g(X,S(Y )) = g(X,Y )λ, and thus if S = λId.
Remark 4.12. The spinorial equation for the class W4 defines a connection with vectorial tor- sion as mentioned in [AF06]. This connection is a G2 connection since the defining spinor ϕ is parallel by definition.
4.4 Adapted connections Let (M 7, g, ϕ) be a 7-dimensional spin manifold with Levi-Civita connection ∇. As usual we identify (3, 0)- and (2, 1)-tensors using g. As in section 3.4 we consider the space Ag of all metric connections and define the map 2 Ξ: End(TM) → Ag,S 7→ Ψ (S., ., .). 3 ϕ The prescription 2 ∇n := ∇ + Sy Ψ 3 ϕ defines a canonical G2-connection (it preserves ϕ), since (Xy Ψϕ)ϕ = −3Xϕ, Ψϕϕ = 7ϕ and ∗ ∗ ∗ Ψϕϕ = −ϕ , ∀ϕ ⊥ ϕ(see Lemma 4.2): 1 ∇n ϕ = ∇ ϕ + Ψ (S(X), ., .) · ϕ = S(X) · ϕ − S(X) · ϕ = 0. X X 3 ϕ The endomorphism S encodes the intrinsic torsion. Abiding by Cartan’s formalism, the set of all metric connections is isomorphic to (compare description of R7,Λ3(R7) and T in Section 3.4) R7 ⊕ R ⊕ R7 ⊕ 2R7 ⊕ ⊕ 2R7 ⊕ R64 |{z} (| {z S0 }) |(g2 S0{z }) 1R7 Λ Λ3R7 T under G2, and this allows us to see 5. SPIN(7) STRUCTURES 29
n Proposition 4.13. The four pure kinds of a G2-manifold correspond to ∇ living in:
n 3 n W1 ⇐⇒ ∇ ∈ Λ W2 ⇐⇒ ∇ ∈ T n 3 n 1 3 W3 ⇐⇒ ∇ ∈ Λ ⊕ T W4 ⇐⇒ ∇ ∈ Λ ⊕ Λ .
Proof. Since for A ∈ G2 we have
−1 −1 −1 A SA 7→ Ψϕ(A SA., ., .) = Ψϕ(AA SA., A., A.) = Ψϕ(SA., A., A.) and Ξ is G2 equivariant. Comparing the dimensions of the corresponding modules, in the cases n W1 and W2 the connection ∇ must have skew symmetric torsion and cyclic traceless torsion. 3 7 Algebraic computations show that for S ∈ W3 the Λ (R ) part does not vanish,
X,Y,Z 1 3 7 0 ≠ S Ψϕ(SX,Y,Z) ∈ Λ (R ) 3 and neither does the T part
1 X,Y,Z 0 ≠ Ψϕ(X,Y,Z) − S Ψϕ(SX,Y,Z) ∈ T . 3
For S ∈ W4 we have S(X) = V × X for some vector V and thus with Lemma 4.4 we get
Ψϕ(SX,Y,Z) = g(V × X,Y × Z) = g(V,Y )g(X,Z) − g(V,Z)g(X,Y ) − ∗Ψϕ(V,X,Y,Z), which is contained in R7 ⊕ Λ3(R7).
The connection of pure type T of the W2 case is discussed in greater detail in [CI07]. Now among all G2-connections, ensuing from the proposition above, there exists a unique connection ∇c with skew-symmetric torsion. Therefore we may write
S = λId + S3 + S4 ∈ W1 + W3 + W4 ∈ 2 7 y with S3 S0 (TM ) and S4 = V Ψϕ for some vector V . 7 Theorem 4.14. Let (M , g, ϕ) be a Riemannian G2 manifold of type W134. The characteristic torsion reads XYZ c 1 T (X,Y,Z) = − S Ψϕ((2λId + 9S3 + 3S4)X,Y,Z). 3 Proof. Consider the projections
∗ 7 ⊗ ⊥ →κ 3 ∗ 7 →Θ ∗ 7 ⊗ ⊥ T M g2 Λ (T M ) T M g2 XYZ ∑ → 1 → ⊗ y Ψϕ(SX,Y,Z) S Ψϕ(SX,Y,Z),T ei (ei T )g⊥ 3 i 2 A little computation shows that the composite Θ ◦ κ is the identity map with eigenvalues 1, 0, 2/9, 2/3 on the four respective summands Wi. But from [FI02] we know that if −2Γ = Θ(T ) for some 3-form T , then T is the characteristic torsion.
5 Spin(7) structures
In a similar spirit as in the case of the other structures, an 8-dimensional oriented Riemannian manifold (M, g) is called a Spin(7) manifold if it has a reduction to Spin(7) ⊂ SO(8), and this is equivalent to the choice of a 4-form Φ which, in a local frame e1, . . . , e8, can be written as
Φ = ϕ + ∗ϕ, and ϕ = e1278 + e3478 + e5678 + e2468 − e2358 − e1458 − e1368. 30 CHAPTER I. G STRUCTURES AND THEIR CHARACTERISTIC CONNECTIONS
We define a skew symmetric endomorphism P (X,Y,.) on TM via
g(P (X,Y,Z),V ) = Φ(X,Y,Z,V ).
∧ ∧ ∧ ∧ We extend the metric g to 3-forms on TM in the usual∑ way, i. e. g(W1 W2 ∑W3,V1 V2 V3) = det(g(Wi,Vi)) for Vi,Wj ∈ TM. For 3-forms ξ = ξijkeijk and η = ηijkeijk let η(ξ) i Hypersurfaces, cones and generalized Killing spinors In this chapter we want to focus on hypersurfaces M ⊂ M¯ . We always consider a G structure on M, which corresponds to a G¯ structure on M¯ . We are able to calculate the corresponding classes. In the first section we look at 6 dimensional M where G = SU(3) and G¯ = G2 on M¯ . Since in this case we have spinorial characterisations of both structures, we are able to do the calculations for a general hypersurface to then restrict ourself to the case of a twisted cone over an SU(3) manifold. For the two cases of almost contact structures and G2 structures on M (corresponding to almost hermitian and Spin(7) structures on M¯ ) we have to choose another technique, which we can only use on (twisted) cones. So in Section 2 we introduce this technique to use it in Sections 3 (the almost contact case) and 4 (the G2 case). In the back of any section we will discuss (generalized) Killing spinors with torsion. Therefore we consider a G structure with Levi-Civita connection ∇, such that there exists a characteristic connection with torsion T . We define the one-parameter family of metric connections ∇s := ∇ + 2sT and recall that ϕ∗ is called a generalized Killing spinor with torsion (gKS) if ∇s ∗ · ∗ X ϕ = A(X) ϕ for some symmetric endomorphism A. In the case where A = λId is a multiple of the identity, this 1 is the definition of a Killing spinor with torsion. The case s = 4 corresponds to the characteristic connection; however, there are many geometric situations in which the Killing equation holds for ̸ n−1 values s = 1/4. Especially the equation for s = 4(n−3) giving the limiting case of the eigenvalue inequality for the Dirac operator with torsion (see [ABBK13]) is interesting. If additionally we have s = 0 we know that for λ real this is the definition of a real Killing spinor and λ is constant. Such spinors realize the equality case of the inequality for the eigenvalue of the Dirac operator (see [Fr80]) and in dimensions 6 and 7 this spinors correspond to nearly K¨ahler structures and nearly parallel G2 structures (see Chapter I). If s = 0 and A is arbitrary symmetric, this is the equation of a so called generalized Killing spinor (see [BGM05] and [FK01]). We saw in Chapter I that such spinors give half flat and cocalibrated structures in dimensions 6 and 7. As the Weingarten map of a hypersurface M ⊂ M¯ is symmetric, the hypersurface theory can be used to construct generalized Killing spinors with (and without, being the case where s = 0) torsion. 31 32 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS 1 SU(3) hypersurfaces in G2 manifolds 7 6 Let (M¯ , g,¯ ϕ) be a 7-dimensional G2 manifold and M a hypersurface with transverse unit direction V T M¯ 7 = TM 6 ⊕ ⟨V ⟩ . (II.1) By restriction the spin bundle Σ¯ of M¯ 7 gives a Spin(6)-bundle Σ over M 6, and similarly the Clifford multiplication · of M 6 is X · ϕ = V Xϕ in terms of the one on M¯ 7 (whose symbol we suppress, as usual). In particular, this implies that any σ ∈ Λ2kM 6 ⊂ Λ2kM¯ 7 of even degree will satisfy σ · ϕ = σϕ. This notation was also used in [BGM05] to describe almost Killing spinors (compare Section 1.2 in this chapter). The second fundamental form g(W (X),Y ) of the immersion (W is the Weingarten map) accounts for the difference between the two Riemannian structures, and in Σ¯ we can compare 1 ∇¯ ϕ = ∇ ϕ − V W (X)ϕ, X X 2 where ∇ and ∇¯ are the Levi-Civita connections of M 6 and M¯ 7 A global spinor ϕ on M¯ 7 (a 6 G2-structure) restricts to a spinor ϕ on M (an SU(3)-structure). The next lemma explains how both the almost complex structure and the spin structure are – in practice – induced by ϕ and the normal V . Lemma 1.1. For any sections ϕ∗ ∈ Σ and vectors X ∈ TM 6 i) V ϕ∗ = j(ϕ∗) ii) V Xϕ = (JϕX)ϕ ∗ ∗ ∗ Proof. The volume form σ7 satisfies σ7ϕ = −ϕ for any spinor ϕ ∈ Σ. Therefore V j(Xϕ) = σ7(Xϕ) = −Xϕ. 6 Another way of interpreting the structure on M is to say that the G2-form Ψϕ defines J by V y Ψϕ = −ω. 7 Lemma 1.2. With respect to decomposition (II.1) the G2-endomorphism of M¯ has the form 1 JϕS − JϕW ∗ S¯ = 2 (II.2) η ∗∗ 6 where (S, η) are the intrinsic tensors of M , Jϕ the almost complex structure, W the Weingarten map. This result was first proved in [CS06] using Cartan-K¨ahlertheory. Our alternative argument is much simpler: Proof. From the definition ∇X ϕ = VS(X)ϕ + η(X)V ϕ and Lemma 1.1 we get ∇¯ X ϕ = ∇X ϕ − 1 − 1 2 V W (X)ϕ = JϕS(X)ϕ 2 JϕW (X)ϕ + η(X)V ϕ. The starred terms in matrix (II.2) should point to the half-obvious fact that the derivative ∇V ϕ cannot be reconstructed from S and η. As a matter of fact, later we will need to know that the bottom row of S¯ is controlled by the product (∇ϕ, V ϕ), so that the entry ∗∗ vanishes when ∇V ϕ = 0. 1. SU(3) HYPERSURFACES IN G2 MANIFOLDS 33 Now we are ready for the main theorems, which explain how to go from M 6 to M¯ 7 (Theorem 1.4) and backwards (Theorem 1.5). The run-up to those requires the following preparatory definition. Recall that the map S is symmetric if the SU(3) structure is of type χ1¯23¯ (see Lemma 3.6, Chapter I). Definition 1.3. The symmetry of the Weingarten endomorphism W expresses half-flatness, i.e. 6 ⊂ ¯ 7 class χ123, by [CS06]. Motivated by that we shall say that a hypersurface M M has (0) type zero if W is the trivial map (meaning ∇¯ = ∇), (I) type one if W is of class χ1¯, (II) type two if W is of class χ2¯, (III) type three if W is of class χ3. Due to the freedom in choosing entries in (II.2), we will take the easiest option (probably also the most meaningful one, geometrically speaking) and consider embeddings where ∇V ϕ = 0. 6 7 Theorem 1.4. Embed (M , g, ϕ) in some (M¯ , g,¯ ϕ) as in (II.1), and suppose the G2-structure to be parallel in the normal direction: ∇¯ V ϕ = 0. 7 6 Then the classes Wα of (M¯ , g,¯ ϕ) depend on the column position (the class of M ) and the row position (the Weingarten type of M 6) as in the table + − + − χ1 χ1 χ2 χ2 χ3 χ4 χ5 0 W13 W4 W3 W2 W3 W2 W234 I W134 W4 W34 W24 W34 W24 W234 II W123 W24 W23 W2 W23 W2 W234 III W13 W34 W3 W23 W3 W23 W234 ( ) Proof. Let A be an endomorphism of R6 and θ a covector. Then the endomorphism A¯ = JϕA 0 θ 0 7 of R is of type W4 if and only if θ = 0 and A is a multiple of the identity, since Jϕ is given by 1 g(X,JϕY ) = 2 Ψϕ(V,X,Y ). ( ) With other similar and easy implications we show that the type of A¯ = JϕA 0 is determined θ 0 by the class of the intrinsic tensors (A, θ) on M 6 in the following way: ∈ − + − (A, θ) χ1 χ1 χ2 χ2 χ3 χ4 χ5 ∈ + − + (JϕA, θ) χ1¯ χ1 χ2 χ2 χ3 χ4 χ5 A¯ ∈ W13 W4 W3 W2 W3 W2 W234 Now the theorem can be proved::( Consider) for example an SU(3) structure (S, η) of type χ3 on a JϕS 0 hypersurface of type one. Then is of type W3 and since W is a multiple of the identity, ( ) η 0 ( ) J S− 1 J W 0 JϕW 0 W ¯ ϕ 2 ϕ 0 0 is of type 4. This immediately states that S = η 0 and thus the type of the G2 structure is W34. We will now do the opposite: start from the ambient space (M¯ 7, g,¯ ϕ) and infer the structure of its codimension-one submanifolds M 6. By inverting formula (II.2) we immediately see from 1 1 S¯| 6 = JS + JW implying J S¯| 6 = −S − W TM 2 ϕ TM 2 34 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS that 1 J S¯| 6 + W = −S ϕ TM 2 and thus 1 S = −J S¯ + W, η(X) = g(SX,V¯ ) ϕ TM 6 2 for any X ∈ TM 6. To conclude, we can state the final result on hypersurfaces (which can be found, in a different form, in [C06], Section 4). 7 Theorem 1.5. Let (M¯ , g,¯ ϕ) be a Riemannian spin manifold of class Wα. Then a hypersurface M 6 orthogonal to V for some V ∈ T M¯ 7 has an induced spin structure ϕ+: its class is an entry in the matrix below that is determined by its column (the Weingarten type) and row position (Wα) W1 W2 W3 W4 0 χ1 χ1245 χ1235 χ145 I χ11 χ1245 χ11235 χ145 II χ12 χ1245 χ12235 χ1245 III χ13 χ12345 χ1235 χ1345 ( ) ∗ Proof. To proceed as in Theorem 1.4, we prove that the class of an endomorphism A¯ = JϕA θ ∗ on R7 determines the class of (A, θ) on a R6 in the following way: A¯ ∈ W1 W2 W3 W4 ∈ (A, θ) χ1 χ1245 χ1235 χ145 If A¯ ∈ W1 we have A¯ = λId and thus η = 0 and A = λJϕ. ¯ ∈ W If A 2 then JϕA is skew-symmetric,( and A has) type χ124. If A¯ is of type W we have S¯ = JϕA η for some symmetric J A. Therefore JA is of 3 η −tr(JϕA) ϕ type χ123 implying the type χ123 for A. Suppose A¯ ∈ W4, so there is a vector Z such that g(X, AY¯ ) = Ψϕ(Z,X,Y ), whence (XY Zϕ, ϕ) = (AY¯ ϕ, Xϕ) 7 6 6 for every X,Y ∈ R . Restrict this equation to X,Y ∈ R and put Z = λV + Z1,Z1 ∈ R . Then JϕA = λJϕ + A1 with (XYZ1ϕ, ϕ) = (A1Y ϕ, Xϕ). Since A1 is skew we have g(X,A1JϕY ) = (Z1XJϕY ϕ, ϕ) = (Z1XV Y ϕ, ϕ) = −(Z1Y V Xϕ, ϕ) = −(Z1YJϕXϕ, ϕ) = −g(Y,A1JϕX) = −g(X,JϕA1Y ), so A1Jϕ = −JϕA1 and A1 has type χ4. Thus JϕA ∈ χ14. From this table it becomes clear that we cannot have a W1-manifold if the derivative of ϕ along V vanishes. Moreover, in case ∇V ϕ = 0 the χ5-component disappears from everywhere, simplifying the matter a little. 1. SU(3) HYPERSURFACES IN G2 MANIFOLDS 35 1.1 Spin cones As usual, start with (M 6, g, ϕ) with intrinsic torsion (S, η). Choose a complex-valued function 1 h = h1 + ih2 : I → S of unit norm defined on some real interval I and set ϕt := h(t)ϕ := h1(t)ϕ + h2(t)j(ϕ) 6 yielding a new (family of) SU(3) structures on M depending on t ∈ I. Then j(ϕ)t = j(ϕt) = j(h1(t)ϕ + h2(t)j(ϕ)) = −h2(t)ϕ + h1(t)j(ϕ) = h(t)j(ϕ). The product of a complex number a ∈ C with an endomorphism A ∈ End(TM) is defined as aA = (Re a)A + (Im a)JϕA. With this definitions we get h(A(X)ϕ∗) = (hA)(X)ϕ∗ = A(X)hϕ¯ ∗ for any spinor ϕ∗. The first observation 6 2 is that the intrinsic torsion of (M , g, ϕt) is given by (h S, η) (cf. (ii) in Remark 3.18, with f = h is constant on M 6), for ∇X ϕt = h∇X ϕ = h(S(X) · ϕ) + hη(X)j(ϕ) ¯ 2 = (hS)(X) · (hhϕ) + η(X)j(ϕ)t = (h S)(X) · ϕt + η(X)j(ϕ)t. Now let us conformally rescale the metric by some positive function f : I → R+ and consider 6 6 2 Mt := (M , f(t) g, ϕt). 6 6 Note that M and Mt have one Levi-Civita connection and one spinor bundle Σ, but distinct · · · ∗ 1 · ∗ ∀ ∗ ∇ Clifford multiplications , t , albeit related by X ϕ = f(t) X t ϕ , ϕ . Because X ϕt = 2 · h2 · 6 h S(X) ϕt + η(X)j(ϕ)t = f S(X) t ϕt + η(X)j(ϕ)t the intrinsic torsion of Mt gets rescaled as h2 ( f S, η). Definition 1.6. The metric cone (M¯ 7, g¯) = (M 6 × I, f 2(t)g + dt2) ¯ 6 equipped with spin structure ϕ := ϕt will be referred to as the spin cone over M . The Levi-Civita connection ∇¯ t of the cone reads f ′(t) ∇¯ Y = ∇ Y − g¯(X,Y )∂ X X f(t) t ′ ∈ 6 − f for X,Y TM , whence the Weingarten map is W = f Id. Computing h′ h′ ∇¯ ϕ¯ = ∇¯ hϕ = h′ϕ = −ih′j(ϕ) = −i hV ϕ = −i V ϕ¯ ∂t ∂t h h we get the intrinsic torsion of M¯ 7 by ′ h2 f JϕS + Jϕ 0 S¯ = f 2f − h′ η i h By decomposing S = λJϕ + µId + R ∈ χ1 ⊕ χ1¯ ⊕ χ2345 the upper-left term can be written as −λIm h2 + µRe h2 + f ′/2 −λRe h2 − µIm h2 Re h2 −Im h2 J + Id + J R + R. f ϕ f f ϕ f 7 ′ Suppose we require M¯ to be of type W1: since S¯ then is a multiple of the identity, we need h /h to be constant, so h(t) = exp(i(ct + d)), c, d ∈ R. Let us see what happens for specific choices of 36 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS structure on the hypersurface. 6 − 1 The sine cone. Start with an SU(3) manifold (M , g, ϕ) of type χ1¯ with S = 2 Id. The choice h = eit/2 produces a cone ( M 6 × (0, π), sin(t)2g + dt2, eit/2ϕ ). ¯ 1 where S = 2 Id. This construction was considered in [FIVU08] and [St09], among others. Other cones of pure type. To obtain different-type structures we start this time by fixing the function h = 1, so that ϕ¯ = ϕ and 1 ′ µ+ 2 f λ 1 Jϕ − Id + JϕR 0 S¯ = f f f . η 0 By selecting different SU(3) structures we have 6 a) Take M to be χ124, say S = µId + R, and µ < 0 constant: the cone 6 2 2 2 ( M × R+, 4µ t g + dt , ϕ ) ( ) 1 − JϕR 0 has S¯ = 2µt , and thus carries a calibrated G2-structure. 0 0 6 b) On M of type χ123 with µ < 0 constant we can build the same cone, but now the resulting type will be W3. ( ) k(t)J 0 c) Take a χ¯-manifold (S = µId). Since ϕ is of type W irrespective of the map k(t), the 1 0 0 4 cone ( M 6 × I, f(t)2g + dt2, ϕ ) is always W4 if we start from χ1¯, since R and λ vanish in this case. When µ < 0 the special choice f(t) = −2µt will additionally give S¯ = 0. This Ansatz is used in [B¨a93]to describe real Killing spinors on Riemannian manifold M 6. 1.2 Killing spinors with torsion 7 c Let (M¯ , g,¯ ϕ) be a G2 manifold with characteristic connection ∇¯ and torsion T¯, and suppose (M 6, g, ϕ) is a submanifold of type one or three, such that V y T¯ = 0, cf. (II.1). The latter equation warrens that T¯ can be restricted to a 3-form on M 6. We decompose the Weingarten map W = µId + W3 with JW3 = −W3J and prove Lemma 1.7. The differential form XYZ L(X,Y,Z) := − S ψϕ(W3(X),Y,Z) − µψϕ(X,Y,Z) satisfies (Xy L)ϕ = −2W (X)ϕ. ∑ − y ⊗ y Proof. In an arbitrary basis e1, .., e6 the torsion is i(ei T )su(3)⊥ ei = 2Γ, where (ei T )su(3)⊥ ⊥ denotes the projection of eiyT by so(6) → su(3) . It is not hard to see that T ∗M 6 ⊗ su(3)⊥ →κ Λ3(T ∗M 6) →Θ T ∗M 6 ⊗ su(3)⊥ 2 1 2 ∑ Sy ψϕ − η ⊗ ω 7→ S (Sy ψϕ − η ⊗ ω); T 7→ ei ⊗ (eiy T ) ⊥ 3 3 3 su(3) i 1. SU(3) HYPERSURFACES IN G2 MANIFOLDS 37 ◦ 1 ◦ satisfy Θ κ|χ3 = 3 Idχ3 and Θ κ|χ1 = Idχ1 . But since SU(3) is the stabilizer of ϕ for any 3-form R ∈ Λ3(T ∗M 6) we have R(X) · ϕ = Θ(R)(X) · ϕ, so (Xy L) · ϕ = Xy(L) · ϕ = Xy(Θ ◦ κ(−ψϕ(3W3 + µId., ., .))) · ϕ = Xy(−ψϕ(W3 + µId., ., .)) · ϕ = −ψϕ(W (X), ., .) · ϕ = −2W (X) · ϕ, proving the lemma. For X ∈ TM 6 we have ∇¯ c ∇¯ 1 y ¯ 0 = X ϕ = X ϕ + 4 (X T )ϕ ∇ 1 y ¯ · − 1 · = X ϕ + 4 (X T ) ϕ 2 W (X) ϕ. ¯ c 6 So if we define T := T|M 6 + L then ∇ := ∇ + T is characteristic on (M , g, ϕ). Hence if there exist characteristic connections on M¯ 7 and M 6, their difference must be L. 7 Note that the restriction ∇¯ V ϕ = 0 on M¯ is not strong enough, since this only implies (V y T¯)ϕ = 0 and not V y T¯ = 0. In the beginning of this chapter we introduced generalized Killing spinor (gKS) ϕ∗ ∇sϕ∗ = A(X) · ϕ∗ for some symmetric endomorphism A : TM 6 → TM 6, where ∇s := ∇ + 2sT . Starting with any G2 manifold of type W13, the corresponding spinor is a generalized Killing spinor (without torsion), since it satisfies ∇X ϕ = S(X)ϕ with symmetric S (see Lemma 4.7 in Chapter I). Examples may arise from 3-Sasaki manifolds as discussed in [AF10]. Suppose the restriction to M 6 of a spinor ϕ∗ is a gKS. Then at any point of M 6 ∇¯ s ∗ ∇¯ ∗ y ¯ ∗ ∇ ∗ y ¯ ∗ − 1 ∗ X ϕ = X ϕ + s(X T )ϕ = X ϕ + s(X T )ϕ 2 V W (X)ϕ ∇s ∗ y ¯ − ∗ − 1 ∗ = X ϕ + s(X (T T ))ϕ 2 V W (X)ϕ − 1 ∗ y ∗ = V (A 2 W )(X)ϕ + s(X L)ϕ . 1 ∇¯ s ∗ y ∗ Choosing A = 2 W annihilates one term so that X ϕ = s(X L)ϕ . Conversely, any ∇¯ s-parallel spinor on M¯ 7 satisfies ∇¯ s ∗ ∇s ∗ y ¯ − · ∗ − 1 · ∗ 0 = X ϕ = X ϕ + s(X (T T )) ϕ 2 W (X) ϕ . To summarise, 7 c Theorem 1.8. Let (M¯ , g,¯ ϕ) be a G2 manifold with characteristic connection ∇¯ and torsion T¯. Let M 6 ⊂ M¯ 7 be a hypersurface, of type one or three, such that V y T¯ = 0. Then (M 6, g = g¯|TM 6 , ϕ) is an SU(3) manifold and i) the characteristic connection of M 6 is ∇ + T¯ + L. ∗ ¯ 7 ∇s ∗ 1 · ∗ 6 ii) Any solution ϕ on M to the gKS equation X ϕ = 2 W (X) ϕ on M must satisfy ∇¯ s ∗ y ∗ X ϕ = s(X L)ϕ . iii) Vice versa, if ϕ∗ is ∇¯ s-parallel on M¯ 7 then it solves 1 ∇s ϕ∗ = −sXy (T¯ − T ) · ϕ∗ + W (X) · ϕ∗. X 2 38 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Example 1.9. Given (M 6, g) we build the twisted cone (M¯ 7 := M 6 × R, g¯ := a2t2g + dt2) 6 ∼ 6 × { 1 } ⊂ ¯ 7 for some a > 0. From the submanifold M = M a M we can only compute the ¯ 7 6 × { 1 } Clifford multiplication of M in the points of M a . As in section 1.1 we therefore consider 6 6 2 2 ∼ 6 × { t } ⊂ ¯ 7 6 Mt := (M , a t g) = Mt a M as a hypersurface. At any point in Mt the spinor bundle 6 ¯ 7 6 of Mt and of M are the same and they can be identified with the spinor bundle of M . We · ∗ 1 ∗ 6 6 get X ϕ = at ∂tXϕ . Since the metric of Mt is just a scaling of the metric of M , their Levi- Civita connections are the same. For the Levi-Civita connection ∇ on M 6 and the Levi-Civita connection ∇¯ on M¯ 7 we have 1 a ∇¯ ϕ∗ = ∇ ϕ∗ + ∂ Xϕ∗ = ∇ ϕ∗ + X · ϕ∗, X X 2t t X 2 − 1 6 as W (X) = t X. Therefore the submanifolds Mt are of type one, and one could determine the possible structures using Theorems 1.4, 1.5. Any 2-form σ on M 6 is a 2-form on M¯ 7 with ∂ty σ = 0, and in addition σ · ϕ∗ = a2t2σϕ∗ for any spinor ϕ∗. 6 7 Let ϕ be an SU(3)-structure on M . We consider the G2-structure on M¯ given by ϕ. Then 2 2 6 c ∂ty Ψϕ = −a t ω. If M has a characteristic connection ∇ with torsion T we have ∇c ∇ 1 y · ∇¯ − a · 1 y · 0 = X ϕ = X ϕ + 4 (X T ) ϕ = X ϕ 2 X ϕ + 4 (X T ) ϕ ∇¯ − a y · 1 y · ∇¯ 1 y − · = X ϕ 4 (X ψϕ) ϕ + 4 (X T ) ϕ = X ϕ + 4 (X (T aψϕ)) ϕ ∇¯ 1 y 2 2 − = X ϕ + 4 (X a t (T aψϕ))ϕ, 2 2 7 showing that T¯ = a t (T − aψϕ) is the characteristic torsion of M¯ . Given a ∇¯ s-parallel spinor ϕ∗ ∇¯ ∗ y ¯ ∗ ∇ ∗ a · ∗ s y ¯ · ∗ 0 = X ϕ + s(X T )ϕ = X ϕ + 2 X ϕ + a2t2 (X T ) ϕ ∇ ∗ a · ∗ y − · ∗ ∇s ∗ a · ∗ − y · ∗ = X ϕ + 2 X ϕ + s(X (T aψϕ)) ϕ = X ϕ + 2 X ϕ as(X ψϕ) ϕ , from which ∇s ∗ − y · ∗ − a · ∗ X ϕ as(X ψϕ) ϕ = 2 X ϕ . ¯ 3 7 Consider the differential form ψϕ ∈ Λ M¯ defined by ¯ 3 3 6 ¯ ψϕ(X,Y,Z) := a t ψϕ(X,Y,Z) for X,Y,Z ∈ TM and ∂tyψϕ = 0 ∇s ∗ − a · ∗ For a Killing spinor solving X ϕ = 2 X ϕ we then have ∇s ∗ a · ∗ ∇ ∗ a · ∗ y · ∗ 0 = X ϕ + 2 X ϕ = X ϕ + 2 X ϕ + s(X T ) ϕ ∗ 2 2 ∗ ∗ 3 2 ∗ ∗ = ∇¯ X ϕ + sa t (Xy T )ϕ = ∇¯ X ϕ + sa t (Xy ψϕ)ϕ + s(Xy T¯)ϕ . Consequently s 0 = ∇¯ s ϕ∗ + (Xy ψ¯ )ϕ∗. X t ϕ Example 1.10. Let (M 7, g, ξ, η, ψ) be a Einstein Sasaki manifold with Killing vector ξ, its dual η and endomorphism ψ such that ψ2 = −id on ξ⊥. Let 1 g := tg + (t2 − t)η ⊗ η, ξ := ξ and η := tη t t t t 2. CONNECTIONS ON CONES AND THE CONE CONSTRUCTION ON SPINORS 39 for t > 0 (compare also Example 3.18). Becker-Bender proved in Lemm 2.18 of [Be12] that 7 gt (M , gt, ξt, ηt, ψ) again is Sasaki. Let furthermore ∇ be the Levi-Civita connection on (M, gt, ξt, ηt, ψ) and T gt the characteristic torsion of the almost contact structure (there exists a characteristic connection since the manifold is Sasaki). Then Becker-Bender proved in Theorem 2.22 of [Be12] the existence of a Killing spinor with torsion for the connection 1 1 ∇gt + ( − )(XyT gt ). X 2t 2 The introduction of so called quasi Killing spinors in [FK00] is a more restricted version of our def- 7 inition of a generalized Killing spinor and leads to a gKS on the Sasaki manifold (M , gt, ξt, ηt, ψ). As stated in [Be12], in this example the gKS and the Killig spinors with torsion are the same. As for a gKS we have ∗ ∗ ∇X ϕ = A(X) · ϕ for some A symmetric, the G2 structure given by this spinor is cocalibrated (of type W13). Remark 1.11. The sign of the Killing constant may be reversed by ϕ∗ j(ϕ∗). The investigation of generalized Killing spinors (with and without torsion) can be pursued in many other contexts. To name but one, the five-dimensional picture was studied by Conti and Salamon [CS07]. 2 Connections on cones and the cone construction on spinors In Section 1 we considered a 6-dimensional hypersurface in a 7-dimensional manifold. As already mentioned in the beginning of this chapter, fortunately in this case, every spinor has the same stabilizer, giving us a correspondence of SU(3) (resp. G2) structures and spinors. This lightens the calculation of the correspondences of SU(3) and G2 structures on hypersurfaces. Unfortu- nately in dimensions higher then 7 the stabilizer of a spinor depends on the spinor. Thus in the more general case of an almost contact structure in dimension 2n + 1 as a hypersurface of an almost hermitian manifold, we are not able to use the technique introduced in Section 1. The same holds for a G2 hypersurface of a Spin(7) manifold. Therefore we introduce a new method for these cases. We restrict ourselves to the case of a twisted cone over a manifold and use a technique of different connections to compare the structures given on a manifold and its cone. This technique was used earlier in several cases, see for example [B¨a93]. 2.1 The cone construction Consider a Riemannian spin manifold (M, g) equipped with a metric connection ∇ with skew symmetric torsion T and connection form ω, meaning that the tensor g(T (X,Y ),Z) = g(∇X Y − ∇Y X − [X,Y ],Z) is skew symmetric. The aim of this Section is to generalize B¨ar’scone con- struction [B¨a93]for Riemannian Killing spinors, i. e. the case when ∇ = ∇g. As an intermediate tool, we define a connection ∇˜ on the spinor bundle by ∇˜ X ϕ = ∇X ϕ + αX · ϕ, with α ∈ R\{0}. Denote by C(Rn) the Clifford algebra of Rn with respect to the standard negative definite eu- clidean scalar product, and by ∆n the spin module of Spin(n). We consider the Clifford mul- n n n n tiplication for X ∈ R ⊂ C(R ) in ∆n. It is the action of an element of R ⊂ spin(n) ⊕ R = n spin(n + 1) ⊂ C(R ) in ∆n. Let PSO(n)M be the SO(n)-principal bundle of frames, ΣM the spinor bundle and ρn : C(n) → GL(∆n) the representation of the Clifford algebra, i. e. ρ∗|spin(n) is the spin(n) representation. Let PSpin(n)M be the Spin(n)-principal bundle. For a local section 40 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS n h in PSO(n)M, we identify TM and PSO(n)M ×SO(n) R via X = [h, η(dh(X))], where η is the solder form. The affine connection ∇˜ induces a connection in the Spin(n + 1)-principal bundle PSpin(n)M ×Spin(n) Spin(n + 1) as follows. Let Φ: PSpin(n)M → PSO(n), θ : Spin(n) → SO(n) ∼ n be the usual projections. We look at spin(n + 1) = spin(n) ⊕ R ⊂ C(n), the restriction of ρ∗ to spin(n + 1), and obtain for a local section k in PSpin(n)M with Φ(k) = h and ΣM ∋ ϕ = [k, σ], ∇˜ X [k, σ] = ∇X [k, σ] + α · [h, η(dhX)] · [k, σ] −1 = [k, dσ(X) + ρ∗(θ∗ (ω(dhX)) + αη(dhX)))σ]. ∗ −1 Thus we get the spin(n + 1)-valued 1-formω ˆ := Φ (θ∗ ω + αη) on PSpin(n)M. We extendω ˆ to n PSpin(n+1)M as follows: For b ∈ PSpin(n)M we have TbPSpin(n+1)M = TbPSpin(n)M ⊕ dLb(R ), where Lb : Spin(n + 1) → PSpin(n+1)M, g 7→ b · g and define n ωˆ(dLbY ) := Y ∈ R ⊂ spin(n + 1). For any b ∈ PSpin(n)M we further extendω ˆ in a Spin(n + 1) equivariant way. One checks that the given form is a connection form. It is the connection form of the connection given by ∇˜ . As in [B¨a93],we consider the SO(n + 1)-principal bundle PSO(n+1)M := PSO(n)M ×SO(n) SO(n + 1) −1 ∗ and calculate the corresponding connection formω ˜ given by θ∗ Φ ω˜ =ω ˆ for the projections Φ: PSpin(n+1)M → PSO(n+1)M and θ : Spin(n + 1) → SO(n + 1) and get ω −2αη ω˜ = . 2αηt 0 We now consider the cone (M,¯ g¯) = (M × R+, a2r2g + dr2) for some fixed a > 0 with principal ¯ ¯ g¯ g¯ SO(n)-bundle of frames PSO(n+1)M, Levi-Civita connection ∇ with connection formω ¯ and projection π : M¯ → M. For simplicity, we will write X ∈ TM for a lift to M¯ of a vector field on M. We define a tensor T¯ on M¯ from the torsion tensor T of ∇ via T¯(X,Y ) := T (X,Y ) for X,Y ⊥ ∂r, ∂ryT¯ = 0. We will notationally not distinguish between the (2, 1) torsion tensors and the corresponding skew symmetric (3, 0)-tensors obtained via g(T (X,Y ),Z). For the metrics g, g¯ on M and M¯ , we 2 2 have a r T (X,Y,Z) = T¯(X,Y,Z) for X,Y,Z ⊥ ∂r. From T¯, we define on M¯ the connection 1 ∇¯ := ∇g¯ + T¯, 2 whose connection form isω ¯. For p ∈ M and s ∈ R+, the tangent bundle of M¯ splits into ¯ ¯ ¯ T(p,s)M = TpM ⊕ R, where dπ(T M) = TM. Thus, for X ∈ TM ⊂ T M, we will write ”X” instead of ”dπX”. With a local orthonormal frame (X1,...,Xn) of M we have an isomorphism of the last two vector bundles given by (Y ∈ Rn+1) 1 1 ψ : π∗(P˜ M) × Rn+1 → T M,¯ [(X , .., X , ∂ ),Y ] 7→ [( X , .., X , ∂ ),Y ]. SO(n+1) SO(n+1) 1 n r ar 1 ar n r ∗ ˜ Thus we can view the connectionω ¯ as a connection of π (PSO(n+1)M), which we again callω ¯. We summarize the different principal bundles with corresponding connections and vector bundles in the following table: 2. CONNECTIONS ON CONES AND THE CONE CONSTRUCTION ON SPINORS 41 bundle connection form vector bundle manifold PSO(n)M ω T M M ˜ PSO(n+1)M ω˜ M ∗ ˜ ∗ ∗ ˜ n+1 ¯ π (PSO(n+1)M) π ω˜ π (PSO(n+1)M) ×SO(n+1) R M ¯ ¯ ¯ PSO(n+1)M ω¯ T M M ∗ ˜ ¯ To determineω ¯ for a local frame h := (X1, .., Xn, ∂r) in π (PSO(n+1)M), X ∈ T M, we need to ∗ ˜ n+1 compute (Y ∈ π (PSO(n+1)M) ×SO(n+1) R ) −1 ψ (∇¯ X ψ(Y )) = [h, d(η(dhY ))(X) +ω ¯(dhX)η(dhY )]. ˜ 1 1 ∈ ⊂ ∗ ˜ × Let h := ( ar X1, .., ar Xn, ∂r) be a local frame in PSO(n+1). For Y TM π (PSO(n+1)M) SO(n+1) n+1 t R we locally have Y = [h, (Y1, .., Yn, 0) ] for functions Yi : M → R and thus ψ(Y ) = ˜ t [h, (Y1, .., Yn, 0) ]. Therefore arψ(Y ) is independent of r and thus a lift of a vector field on M. Using the O’Neill formulas [O’N83, p. 206], we compute for lifts X,Y of vector fields in TM and the Levi-Civita connection ∇¯ g¯ of M¯ g¯ g¯ g¯ 1 g¯ g 1 ∇¯ ∂r = 0, ∇¯ X = ∇¯ ∂r = X, ∇¯ Y = ∇ Y − g¯(X,Y )∂r. ∂r ∂r X r X X r Adding the torsion tensor T¯, this implies 1 1 ∇¯ ∂ = 0, ∇¯ X = ∇¯ ∂ = X, ∇¯ Y = ∇ Y − g¯(X,Y )∂ . ∂r r ∂r X r r X X r r ¯ ∗ ˜ n+1 For X ∈ T M and Y ∈ TM ⊂ π (PSO(n+1)M) ×SO(n+1) R we have −1 ∇¯ −1 ∇¯ ! t t t ψ ( ∂r ψ(∂r)) = ψ ( ∂r ∂r) = 0 = [h, d((0..0, 1) )(∂r)+¯ω(dh∂r)(0..0, 1) ] = [h, ω¯(dh∂r)(0..0, 1) ] and 1 1 1 ψ−1(∇¯ ψ(Y )) = ψ−1(∇¯ arψ(Y )) = ψ−1( ∇¯ arψ(Y ) + (∂ )arψ(Y )) ∂r ∂r ar ar ∂r r ar 1 1 1 = ψ−1( (arψ(Y )) − arψ(Y )) = 0 ar r ar2 ! t = [h, 0 +ω ¯(dh∂r)(Y1, .., Yn, 0) ] ˜ t ˜ and thusω ¯(dh∂r) = 0. Furthermore X = [h, ar(X1, .., Xn, 0) ] = [h, arη(dhX)] and we get 1 ψ−1(∇¯ ψ(∂ )) = ψ−1(∇¯ ∂ ) = ψ−1( X) = ψ−1([h,˜ aη(dhX)]) = [h, aη(dhX)], X r X r r ˜ t 2 2 t proving aη =ω ¯ · ∂r. Since ψ(Y ) = [h, (Y1, .., Yn, 0) ], we haveg ¯(X, arψ(Y )) = a r η(dhX) · t (Y1, .., Yn) . Furthermore we have ˜ t t t ∇X arψ(Y ) = [h, ar(d(Y1, ..Yn, 0) (X) + ar(ω(dhX)(Y1, .., Yn) , 0) ] and obtain 1 1 1 1 ψ−1(∇¯ ψ(Y )) = ψ−1( ∇¯ arψ(Y )) = ψ−1( ∇ arψ(Y ) − g¯(X, arψ(Y ))∂ ) X ar X ar X ar r r −1 ˜ t t t t t t = ψ ([h, d(Y1, .., Yn, 0) (X) + (ω(dhX)(Y1, .., Yn) , 0) − aη(dhX) (Y1, .., Yn, 0) (0, .., 0, 1) ]). 42 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Combining all these results yields ω aη ω¯ = . −aηt 0 ¯ ¯ 1 1 − If one changes the orientation of M (a local SO(M) frame is then given by ( ar X1, ..., ar X2, ∂r)), we obtain the alternative connection form ω −aη . aηt 0 For a ∇-Killing spinor on M with real Killing number α, we thus choose the cone constant a = −2α for α < 0 and a = 2α for α > 0. Hence, the cone depends on the Killing number and the construction only makes sense if α ∈ R\{0}, as we had assumed from the beginning. In particular, the results cannot be applied to ∇-parallel spinors (α = 0). The pullback of the connectionω ˜ under the projection π : M¯ → M is the same as the connectionω ¯ on M¯ , thus their holonomy groups Hol(˜ω) and Hol(¯ω) are the same. Since the second Stiefel-Whitney class of M¯ = M × R is given by [Th52, p.142] w2(M¯ ) = w2(M) + w2(R) + w1(M) ⊗ w1(R), we conclude that M¯ is spin, since we assumed M to be spin. Let us now have a closer look at spinors on M and M¯ . A parallel spinor of (M,¯ ω¯) is the same as a trivial factor of the action of the holonomy group Hol(¯ω) = Hol(˜ω) on ∆n+1. A Killing spinor on (M,ω) corresponds to a trivial factor of the action of the same group on the space ∆n. + ⊕ − For n = dim(M) odd, the spin representation splits into ∆n+1 = ∆n ∆n . Changing the ¯ + orientation of M (changing from negative to positive α and vice versa) means interchanging ∆n − ¯ + − and ∆n . Thus, a parallel spinor on M is either in ∆n or in ∆n , giving either a Killing spinor with positive or with negative Killing number α. For n even, we have ∆n = ∆n+1 and, by interchanging the orientation, we obtain for any parallel spinor in M¯ one Killing spinor with positive, and one with negative Killing number α. We summarize these results in the following lemma: Lemma 2.1. For a Riemannian spin manifold (M, g) with metric connection ∇ with skew sym- metric torsion T , consider the manifold (M,¯ g¯) with connection ∇¯ with skew symmetric torsion T¯ as constructed above. The following correspondence holds: • If n = dim(M) is odd, any ∇¯ -parallel spinor on M¯ corresponds to a ∇-Killing spinor on 1 − 1 M, with either positive or negative Killing number 2 a or 2 a. • If n is even, any ∇¯ -parallel spinor on M¯ corresponds to a pair of ∇-Killing spinors on M ± 1 with Killing number 2 a. Remark 2.2. For dim M even, one can write down the bijection between Killing spinors with torsion with Killing numbers ±α explicitly: If ϕ has Killing number α and decomposes into + − ϕ = ϕ+ + ϕ− in the spin bundle ΣM = Σ M ⊕ Σ M, then ϕ+ − ϕ− is a Killing spinor with Killing number −α. This is the same argument as in the Riemannian case [BFGK91, p.121]. Remark 2.3. The careful reader will have noticed that our cone is slightly more general than in [B¨a93],where the computations are done for cone constant a = 1. This stems from the fact that in the Riemannian case, the Killing number is determined through n = dim M and Scalg (remember that the manifold has to be Einstein), hence the cone can be normalized in such a way that a = 1. For our applications, this is too restrictive. 2. CONNECTIONS ON CONES AND THE CONE CONSTRUCTION ON SPINORS 43 2.2 The cone correspondence for spinors Let the cone (M,¯ g¯) over M with Levi-Civita connection ∇¯ g¯ carry a G¯ structure and assume that there is a connection ∇ on M such that its lift ∇¯ to M¯ with torsion T¯ is the characteristic connection on M¯ with respect to the given G¯ structure. Given a G structure on M, we shall construct an induced G¯ structure on M¯ in the following sections. We will see that the characteristic connection ∇c on M (with torsion T c) does not lift to the characteristic connection ∇¯ on M¯ (with torsion T¯, introduced by a connection ∇ on M with torsion T ). In particular the lift T c of the characteristic torsion to M¯ is not the characteristic torsion on M¯ . So the tensor T c − T is not zero and will play an important role in the following. As introduced in the beginning of this section, we will use Killing spinors (not generalized Killing spinors) with torsion ϕ satisfying the equation ∇s X ϕ = αXϕ ∈ R − { } ∇s ∇g c for some Killing number α 0 and some value of s, where X Y = X Y + 2sT (X,Y ). This definition includes the choice that we do not view a parallel spinor (α = 0) as a special case of a Killing spinor. A priori, solutions of this equation with α ∈ C − R are conceivable, but we are not aware of any. In any event, the cone construction would not work for such an α. The connection ∇¯ s on M¯ is then given by ∇¯ s = ∇¯ g¯+2sT¯. We obtain the following correspondence between connections on M¯ and connections on M: Connections on M Connections on M¯ ∇s = ∇g + 2sT c ∇¯ g¯ + 2sT c = ∇¯ s − 2s(T¯ − T c) ∇g + 2sT = ∇s + 2s(T − T c) ∇¯ s = ∇¯ g¯ + 2sT¯ A direct application of Lemma 2.1 implies: Lemma 2.4. For α ∈ R − {0}, we have the following correspondence between spinors on M spinors on M¯ ∇s ∇¯ s − y ¯ − c X ϕ = αXϕ X ϕ sX (T T )ϕ = 0 ∇s y − c ∇¯ s X ϕ + sX (T T )ϕ = αXϕ X ϕ = 0 1 − 1 For dim(M) odd, there is one spinor on M with either α = 2 a or α = 2 a. ± 1 If dim(M) is even, there is a pair of spinors with Killing numbers α = 2 a on M. 1 In particular for s = 4 we obtain the following correspondence: spinors on M spinors on M¯ ∇c ∇¯ 1 y ¯ − c X ϕ = αXϕ X ϕ = 4 X (T T )ϕ ∇c − 1 y − c ∇¯ X ϕ = αXϕ 4 X (T T ) ϕ X ϕ = 0 In the following sections we look at the corresponding structures on M¯ , their classifications and the correspondences of spinors on M and M¯ . 44 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS 3 Almost hermitian cones over almost contact manifolds Let (M, g, ψ, η) be an n-dimensional metric almost contact structure. As in Section 2.1 we construct the twisted cone M¯ over M and define an almost hermitian structure J on M¯ via J(ar∂r) := ξ, J(ξ) := −ar∂r and J(X) = −ψ(X) for X ⊥ ξ, ∂r. The identity ψ2 = −Id + η ⊗ ξ immediately implies J 2 = −Id. Definition 3.1. If M admits a characteristic connection ∇c with skew symmetric torsion T c satisfying ∇cψ = ∇cη = 0, we define a connection ∇ with skew symmetric torsion T c − ∧ ∇ ∇c − ∧ T := T 2aη F and thus X Y = X Y a(η F )(X,Y,.). In particular: If the almost metric contact structure is Sasakian and the Killing number happens to satisfy |α| = 1/2 (like in the Riemannian case), the cone is constructed with a = 1, and thus T c = η ∧ dη = 2aη ∧ F and ∇ = ∇g, the Levi-Civita connection. Thus, ∇ and T measure in some sense the difference to the Riemannian Sasakian case. Although the role of T is clearly exposed in Section 2.2, this is not sufficient to determine T completely. Rather, the formula for T has to be found by trying a suitable Ansatz, the motivation for which comes precisely from the Riemannian case just described. Since T is unique, the definition is justified a posteriori by yielding the desired correspondence. Theorem 3.2. If (M, g, ψ, η) is an almost contact metric structure, (M,¯ g,¯ J) is an almost hermitian manifold. If furthermore M admits a characteristic connection, consider the connection ∇ defined above. Then the appendant connection ∇¯ on M¯ is almost complex, ∇¯ J = 0. Remark 3.3. This shows in particular that ∇¯ is the unique characteristic connection of M¯ with respect to J. Furthermore, the theorem includes the claim that the existence of a characteristic connection for the almost contact metric structure on (M, g, ψ, η) suffices to imply that the induced almost hermitian structure on M¯ does also admit a characteristic connection. We first prove Lemma 3.4. On M, Definition 3.1 implies (∇Y ψ)X = ag(Y,X)ξ − aη(X)Y, (II.3) and we have a) aψ(X) = −∇X ξ, b) ξ is a Killing vector field, g(∇Y ξ, X) = −g(∇X ξ, Y ) and thus its integral curves are geodesics, c) dη = 2aF + ξyT . Proof of Lemma 3.4. Using the definition ∇ = ∇c − aη ∧ F with the equation ∇cψ = 0, we directly compute (∇Y ψ)X = ag(Y,X)ξ − aη(X)Y . Identity (II.3) and ψ(ξ) = 0 imply for X ∈ TM aX − ag(X, ξ)ξ = −(∇X ψ)ξ = ∇X (ψ(ξ)) − (∇X ψ)ξ = ψ(∇X ξ). Since ∇X ξ ⊥ ξ, applying ψ yields aψ(X) = ψ(aX − ag(X, ξ)ξ) = −∇X ξ. 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 45 Since g(X, ψ(Y )) = −g(ψ(X),Y ), we can conclude from equation (II.3) the statement b) of the lemma, which is also a consequence of Theorem 8.2 in [FI02]. For X,Y ∈ TM, we obtain with statement a) dη(X,Y ) = Xη(Y ) − Y η(X) − η([X,Y ]) = Xg(Y, ξ) − Y g(X, ξ) − g([X,Y ], ξ) = g(∇X Y, ξ) + g(Y, ∇X ξ) − g(∇Y X, ξ) − g(X, ∇Y ξ) − g([X,Y ], ξ) = T (X, Y, ξ) − g(Y, aψ(X)) + g(X, aψ(Y )) = T (X, Y, ξ) + 2aF (X,Y ) which finishes the proof. Proof of Theorem 3.2. One easily checks thatg ¯(JX,JY ) =g ¯(X,Y ) for X,Y ∈ T M¯ and thus J is an almost hermitian structure. We have to show ∇¯ J = 0, meaning 0 = ∇¯ Y (J(X)) − J(∇¯ Y X). To do so, we distinguish the following cases: If X ∈ TM, X ⊥ ξ and Y ∈ TM we have 1 ∇¯ (J(X)) − J(∇¯ X) = −∇¯ (ψ(X)) − J(∇ X − g¯(Y,X)∂ ) Y Y Y r r 1 1 = −∇ (ψ(X)) + g¯(Y, ψ(X))∂ − J(∇ X) + g¯(Y,X)ξ Y r r Y ar2 2 = −(∇Y ψ)(X) − ψ(∇Y X) + a rg(Y, ψ(X))∂r − J(∇Y X) + ag(Y,X)ξ. With identity (II.3) and since η(X) = 0, ψ(ξ) = 0 we get 2 ∇¯ Y (J(X)) − J(∇¯ Y X) = −aη(X)Y − ψ(∇Y X) + a rg(Y, ψ(X))∂r − J(∇Y X) = −ψ(∇Y X + ag(Y, ψ(X))ξ) − J(ag(Y, ψ(X))ξ + ∇Y X), which is equal to zero if ∇Y X + ag(Y, ψ(X))ξ is perpendicular to ξ and ∂r. Obviously it is perpendicular to ∂r. We have g(∇Y X + ag(Y, ψ(X))ξ, ξ) = 0 if 0 = g(∇Y X, ξ)+g(Y, aψ(X)) = −g(X, ∇Y ξ)+g(Y, aψ(X)) = g(X, aψ(Y ))+g(Y, aψ(X)) = 0. ∈ ⊥ ∇¯ − ∇¯ 1 − 1 If X TM, X ξ and Y = ∂r we have Y (J(X)) J( Y X) = r J(X) J( r X) = 0. If X = ξ, Y = ∂r we get 1 ∇¯ (J(X)) − J(∇¯ X) = ∇¯ (−ar∂ ) − J( ξ) = −a∂ − ar∇¯ ∂ + a∂ = 0. Y Y ∂r r r r ∂r r r Given X = ξ and Y = ξ we have 1 ∇¯ (J(X)) − J(∇¯ X) = −∇¯ (ar∂ ) − J(∇ ξ − g¯(ξ, ξ)∂ ) = −aξ + aξ = 0. Y Y ξ r ξ r r If X = ξ, Y ∈ TM, Y ⊥ ξ we have 1 ∇¯ (J(X))−J(∇¯ X) = −∇¯ (ar∂ )−J(∇ ξ− g¯(Y, ξ)∂ ) = −aY +J(aψ(Y )) = −aY +aY = 0. Y Y Y r Y r r Given X = ∂r, Y ⊥ ξ, Y ∈ TM we get 1 1 1 1 ∇¯ (J(X)) − J(∇¯ X) = ∇¯ ( ξ) − J( Y ) = − aψ(Y ) − J( Y ) = 0. Y Y Y ar r ar r In the case X = ∂r and Y = ξ we have 1 1 1 1 ∇¯ (J(X)) − J(∇¯ X) = ∂¯ ( ξ) − J( ξ) = ∇ ξ − g¯(ξ, ξ)∂ + a∂ = −a∂ + a∂ = 0. Y Y ξ ar r ar ξ ar2 r r r r ∇¯ 1 − 1 1 ∇¯ The last case is given by X = Y = ∂r. Then we have ∂r ( ar ξ) = ar2 ξ + ar ∂r ξ = 0. 46 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Let (M, g) be a Riemannian manifold such that the above constructed manifold (M,¯ g¯) carries an almost hermitian structure J. We have J(∂r) ⊥ ∂r. We consider the manifold M = M ×{1} ⊂ M¯ and define for X ∈ TM: ξ := aJ(∂r), η(X) := g(X, ξ) and ψ(X) := −J(X) +g ¯(J(X), ∂r)∂r. We get an almost contact structure on M: 2 ψ (X) = − J(−J(X) +g ¯(J(X), ∂r)∂r) +g ¯(J(−J(X) +g ¯(J(X), ∂r)∂r), ∂r)∂r = − X +g ¯(X,J(∂r))J(∂r) = −X + g(X, ξ)ξ = −X + η(X)ξ and 1 g(ψ(X), ψ(Y )) = g¯(−J(X) +g ¯(J(X), ∂ )∂ , −J(Y ) +g ¯(J(Y ), ∂ )∂ ) a2 r r r r 1 = (g¯(J(X),J(Y )) − g¯(X, J∂ )¯g(Y,J(∂ ))) = g(X,Y ) − η(X)η(Y ). a2 r r Conversely to Theorem 3.2 we prove: Theorem 3.5. Consider the manifold M¯ equipped with a connection ∇¯ with skew symmetric torsion T¯ being the lift of a connection ∇ with torsion T on M. If the connection ∇¯ is almost complex on M¯ , we have (∇X ψ)(Y ) = ag(X,Y )ξ−aη(Y )X and thus the characteristic connection ∇c on M = M × {1} has torsion T c = T + 2aη ∧ F . Proof. Given a parallel J on M¯ we have for X,Y ∈ TM (∇X ψ)(Y ) =∇¯ X (ψ(Y )) +g ¯(X, ψ(Y ))∂r − ψ(∇¯ X Y +g ¯(X,Y )∂r) − ∇¯ ∇¯ − = X J(Y ) +| (Xg¯(J(Y{z), ∂r))∂}r +¯g(J(Y ), ∂r) X ∂r g¯(X,J(Y ))∂r g¯(∇¯ X J(Y ),∂r )∂r +g¯(J(Y ),X)∂r + J(∇¯ X Y ) +g ¯(X,Y )J(∂r) − g¯(J(∇¯ X Y ), ∂r)∂r = − g¯(Y,J(∂r))X +g ¯(X,Y )J(∂r) = ag(X,Y )ξ − aη(Y )X. ∇c ∇ 1 c − ∇c ∇c Writing = + 2 (T T ) and using equation (II.3), the condition ψ = η = 0 leads to the following two conditions for X,Y,Z ∈ TM 1 1 (T c − T )(X, ψ(Y ),Z) + (T c − T )(X, Y, ψ(Z)) = ag(X,Y )η(Z) − ag(X,Z)η(Y ) 2 2 and 1 (T c − T )(X, Y, ξ) = (∇ η)(Y ) = g(X, ∇ ξ) = F (Y,X). 2 X Y Defining T c = T +2aη ∧F , T c satisfies these two conditions and thus ∇c is the unique ([AFH13]) characteristic connection of the almost contact metric structure. Remark 3.6. Conversely given an almost complex structure on M and ∂r = ξ, η =g ¯(., ξ) we do not have an almost contact structure on M¯ since dη = 0. From now on we assume that M and M¯ admit an almost contact structure and an almost hermi- tian structure, respectively, both admitting characteristic connections ∇c and ∇¯ as introduced above. 3.1 The classification of metric almost contact structures and the cor- responding classification of almost hermitian structures on the cone We look at the classification of the geometric structures on M¯ and M. We first prove the following two lemmata. 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 47 Lemma 3.7. The Nijenhuis tensor N¯ of the almost hermitian structure on M¯ restricted to TM and the Nijenhuis tensor N of the almost contact structure on M are related via a2r2N = N¯. Furthermore, the following conditions are equivalent: • ∂ryN¯ = 0, • dη(X, ψY ) + dη(ψX, Y ) = 0 on TM, • ξyN = 0. In particular N = 0 if and only if N¯ = 0. Remark 3.8. In [HTY12], T. Houri, H. Takeuchi, and Y. Yasui considered hermitian manifolds M¯ with a vanishing Nijenhuis tensor N¯. They showed that in this case N = 0 and thus M is a normal almost contact manifold, which also is an immediate consequence of Lemma 3.7. Remark 3.9. Since N = 0 if and only if N¯ = 0, the condition N¯ = 0 is sometimes used for the definition of an almost contact metric manifold to be normal (see for example [CG90]). Proof of Lemma 3.7. Since we have 1 1 g¯((∇¯ g¯ J)Y,Z) =g ¯((∇¯ J)Y + (J¯T (X,Y )−T¯(X,JY )),Z) = − (T¯(X,Y,JZ)+T¯(X,JY,Z)), X X 2 2 the Nijenhuis tensor of M¯ is given by ¯ ∇¯ g¯ − ∇¯ g¯ ∇¯ g¯ − ∇¯ g¯ N(X,Y,Z) =g ¯(( X J)(JY ),Z) g¯(( Y J)(JX),Z) +g ¯(( JX J)(Y ),Z) g¯(( JY J)(X),Z) = T¯(X,Y,Z) − T¯(JX,JY,Z) − T¯(JX,Y,JZ) − T¯(X,JY,JZ), whereas the Nijenhuis tensor on M is ∇g − ∇g ∇g − ∇g N(X,Y,Z) = g(( X ψ)(ψ(Y )) ( Y ψ)(ψ(X)) + ( ψ(X)ψ)(Y ) ( ψ(Y )ψ)(X),Z) ∇g − ∇g + η(X)g( Y ξ, Z) η(Y )g( X ξ, Z). Identity (II.3) implies 1 g((∇g ψ)(Y ),Z) = ag(X,Y )η(Z) − ag(X,Z)η(Y ) − (T (X, ψ(Y ),Z) + T (X, Y, ψ(Z))) X 2 and hence we obtain for N(X,Y,Z) = 1 1 ag(X, ψ(Y ))η(Z) − T (X, ψ2(Y ),Z)) − T (X, ψ(Y ), ψ(Z)) 2 2 1 1 − ag(Y, ψ(X))η(Z) + T (Y, ψ2(X),Z)) + T (Y, ψ(X), ψ(Z)) 2 2 1 1 ag(ψ(X),Y )η(Z) − ag(ψ(X),Z)η(Y ) − T (ψ(X), ψ(Y ),Z)) − T (ψ(X), Y, ψ(Z)) 2 2 1 1 − ag(ψ(Y ),X)η(Z) + ag(ψ(Y ),Z)η(X) + T (ψ(Y ), ψ(X),Z)) + T (ψ(Y ), X, ψ(Z)) 2 2 1 1 + η(X)g(∇c ξ, Z) − η(X)T c(Y, ξ, Z) − η(Y )g(∇c ξ, Z) + η(Y )T c(X, ξ, Z), Y 2 X 2 which is the same as 1 1 = T (X,Y,Z) − η(Y )T (X, ξ, Z) − η(X)T (ξ,Y,Z) − T (X, ψ(Y ), ψ(Z)) 2 2 − T (ψ(X), Y, ψ(Z)) − T (ψ(X), ψ(Y ),Z) − aη(Y )g(ψ(X),Z) + aη(X)g(ψ(Y ),Z) 1 1 − η(X)T (Y, ξ, Z) − η(X)aF (Z,Y ) + η(Y )T (X, ξ, Z) + η(Y )aF (Z,X). 2 2 48 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS For X ∈ TM we have J(X) + η(X)ar∂r = J(X − η(X)ξ) = −ψ(X − η(X)ξ) = −ψ(X). Since ∂ryT¯ = 0 for X,Y,Z ∈ TM we get T¯(J(X),Y,Z) = −a2r2T (ψ(X),Y,Z) (II.4) and also T¯(J(X),J(Y ),Z) = a2r2T (ψ(X), ψ(Y ),Z) etc. With this result we have 1 N(X,Y,Z) = (T¯(X,Y,Z) − T¯(JX,JY,Z) − T¯(JX,Y,JZ) − T¯(X,JY,JZ)) a2r2 and thus we get the desired result N¯(X,Z,Z) = a2r2N(X,Y,Z) for X,Y,Z ∈ TM. By definition of the Nijenhuis tensor we have ∂ryN¯ = 0 if and only if for X,Y ∈ TM 0 = T¯(ξ, JX, Y ) + T¯(ξ, X, JY ) ⇐⇒ 0 = T (ξ, ψX, Y ) + T (ξ, X, ψY ). The relations ξyT = dη − 2aF and F (ψX, Y ) + F (X, ψY ) = g(ψX, ψY ) + g(X, ψ2Y ) = 0 imply that ∂ryN¯ = 0 holds if and only if dη(ψX, Y ) + dη(X, ψY ) = 0. In [FI02] the identity N(X, Y, ξ) = dη(X,Y ) − dη(ψX, ψY ) is proved and we get N(ψX, Y, ξ) = dη(ψX, Y ) + dη(X, ψY ) − η(X)dη(ξ, ψY ). The identity ξyT = dη − 2aF implies ξydη = 0 and thus dη(ψX, Y ) + dη(X, ψY ) = 0 if and only if N(ψX, Y, ξ) = 0. Since N is skew symmetric we have N(ξ, Y, ξ) = 0 and thus N(ψX, Y, ξ) = 0 is equivalent to ξyN = 0. Lemma 3.10. For Z ∈ T M¯ let ZM be the projection of Z onto TM. Then we have δω(Z) = −(δF − a(n − 1)η)(ZM ). Proof. For X,Y,Z ∈ T M¯ we have 1 1 (∇¯ g¯ ω)(Y,Z) = (∇¯ ω)(Y,Z) − ω(− T¯(X,Y ),Z) − ω(Y, − T¯(X,Z)) X X 2 2 1 = (T¯(X,JY,Z) + T¯(X,Y,JZ)). 2 { } { 1 1 } For a local ONB e1, .., en = ξ of TM we get the local ONB e¯1 = ar e1, .., e¯n = ar en, e¯n+1 = ∂r of T M¯ . In this basis and for Z ∈ T M¯ we compute − n∑+1 1 n∑1 1 1 1 1 1 δω(Z) = − (∇¯ g¯ ω)(¯e ,Z) = − T¯( e , Je ,Z) − T¯( ξ, −∂ ,Z) − T¯(∂ , J∂ ,Z). e¯i i 2 ar i ar i 2 ar r 2 r r i=1 i=1 Since ∂ryT¯ = 0, with equation (II.4) and the fact that ψ(en) = 0, we have − − 1 n∑1 1 n∑1 δω(Z) = T (e , ψe ,Z ) = (T c(e , ψe ,Z ) − 2a(η ∧ F )(e , ψe ,Z )) 2 i i M 2 i i M i i M i=1 i=1 − 1 n∑1 1 ∑n = (T c(e , ψe ,Z ) − 2aη(Z )F (e , ψe )) = T c(e , ψe ,Z ) + aη(Z )(n − 1) 2 i i M M i i 2 i i M M i=1 i=1 = −(δF − a(n − 1)η)(ZM ), finishing the proof. 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 49 We consider the Gray-Hervella classification [GH80] of almost hermitian structures, given in Section 2 of Chapter I. Since we want to work with characteristic connections, we will only consider structures of class χ1 ⊕ χ3 ⊕ χ4. We first translate the conditions of this classification for the almost hermitian structure on M¯ to conditions of the almost contact structure on M. For the discussion of the classification of almost contact structures and the correspondences to the classification of almost hermitian structures see Theorem 3.12. Theorem 3.11. We have the following correspondence between Gray-Hervella classes of almost hermitian structures on the cone M¯ and defining relations of almost contact metric structures on M: Class of M¯ defining relation on M¯ corresponding relation on M ∇¯ g¯ ∇g K¨ahler J = 0 ( X F )(Y,Z) = aη(Y )g(X,Z) −aη(Z)g(X,Y ) χ3 δω = N¯ = 0 N = 0, δF = a(n − 1)η ∇¯ g¯ −1 ∇g δF (ξ) ( X ω)(Y,Z) = n−1 [g¯(X,Y )δω(Z)( X F )(Y,Z) = n−1 (g(X,Z)η(Y ) χ4 −g¯(X,Z)δω(Y ) − g¯(X,JY )δω(JZ) −g(X,Y )η(Z)) +¯g(X,JZ)δω(JY )] χ1 ⊕ χ3 δω = 0 δF = a(n − 1)η χ3 ⊕ χ4 N¯ = 0 N = 0 Furthermore, a structure on M¯ is never nearly K¨ahler(of class χ1) nor of mixed class χ1 ⊕ χ4. Proof. We have aη(Y )g(X,Z) − aη(Z)g(X,Y ) = aη ∧ F (X, Y, ψZ) + aη ∧ F (X, ψY, Z). K¨ahlercase: Since the characteristic connection on M¯ is unique, we have the following equiva- lences ∇¯ g¯J = 0 ⇔ ∇¯ g¯ = ∇¯ ⇔ T¯ = 0 ⇔ T = 0 ⇔ T c = 2aη ∧ F. For a metric connection ∇˜ with skew symmetric torsion T˜ on M one calculates 1 1 (∇˜ F )(Y,Z) = (∇g F )(Y,Z) − T˜(X, ψY, Z) − T˜(X, Y, ψZ). X X 2 2 c ∧ ∇g ∧ ∧ Thus, T = 2aη F implies ( X F )(Y,Z) = aη F (X, Y, ψZ) + aη F (X, ψY, Z) and conversely ∇g ∧ ∧ the condition ( X F )(Y,Z) = aη F (X, Y, ψZ) + aη F (X, ψY, Z) yields 1 1 (∇˜ F )(Y,Z) = (aη ∧ F − T˜)(X, ψY, Z) + (aη ∧ F − T˜)(X, Y, ψZ). X 2 2 The uniqueness of the characteristic connection ∇c on M thus implies T c = 2aη ∧ F . Case χ3: Consider an almost hermitian structure on M¯ of class χ3 defined by δω = N¯ = 0. With Lemma 3.7 and 3.10 we have N¯ = δω = 0 if and only if N = 0 and δF − a(n − 1)η = 0. Case χ4: The defining relation for the class χ4 of an almost hermitian manifold M¯ −1 (∇¯ g¯ ω)(Y,Z) = [g¯(X,Y )δω(Z) − g¯(X,Z)δω(Y ) − g¯(X,JY )δω(JZ) +g ¯(X,JZ)δω(JY )] X n − 1 50 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS translates with Lemma 3.10 for X,Y,Z ∈ T M¯ into 1 1 T¯(X,Y,JZ) + T¯(X,JY,Z) = 2 2 1 [g¯(X,Y )(δF (Z ) − a(n − 1)η(Z )) − g¯(X,Z)(δF (Y ) − a(n − 1)η(Y )) n − 1 M M M M − g¯(X,JY )(δF ((JZ)M ) − a(n − 1)η((JZ)M )) +g ¯(X,JZ)(δF ((JY )M ) − a(n − 1)η((JY )M ))]. ∈ ¯ − − − 2 2 1 − For X T M we haveg ¯(∂r,JX) = g¯(J∂r,X) = g¯(J∂r,XM ) = a r g( ar ξ,XM ) = arη(XM ) and for X ∈ TM we have (JX)M = −ψX. In the case where X = ∂r and Y,Z ∈ TM, the defining relation is equivalent to 0 = arη(Y )(δF (−ψZ) − a(n − 1)η(−ψZ)) − arη(Z)(δF (−ψY ) − a(n − 1)η(−ψY )). This is satisfied if and only if 0 = (η(Y )δF (ψZ) − η(Z)δF (ψY )) = η ∧ (δF ◦ ψ)(Y,Z). Taking Y = ξ we receive the condition F ◦ ψ = 0, which obviously is sufficient too. If X = Y = ∂r, Z ∈ TM the defining relation leads to 1 n − 1 0 = δF (Z) − a(n − 1)η(Z) − arη(Z)(δF ( ξ) − ), ar r which is the same as 0 = δF (Z) − η(Z)δF (ξ) = −δF (ψ2Z), already being satisfied if δF ◦ ψ = 0. The case Y = Z = ∂r leads to 0 = 0. Given Y = ∂r and X,Z ∈ TM we get 1 1 1 1 T¯(X, ξ, Z) = [arη(X)δF (−ψ(Z)) − a2r2F (X,Z)(δF ( ξ) − a(n − 1) )]. 2ar n − 1 ar ar Since we already have the condition δF ◦ ψ = 0 this is equivalent to 2 dη(X,Z) − 2aF (X,Z) = (ξyT )(X,Z) = F (X,Z)(δF (ξ) − a(n − 1)). n − 1 2 ∈ This is the same as dη = n−1 δF (ξ)F . At last we look at X,Y,Z TM. Again we already have δF ◦ ψ = 0 1 1 − T (X, Y, ψZ) − T (X, ψY, Z) 2 2 1 = [g(X,Y )(δF (Z) − a(n − 1)η(Z)) − g(X,Z)(δF (Y ) − a(n − 1)η(Y ))] n − 1 δF δF = g(X,Y )( − aη)(Z) − g(X,Z)( − aη)(Y ). n − 1 n − 1 Furthermore we have 1 1 1 − (T (X, Y, ψZ) + T (X, ψY, Z)) = − T c(X, Y, ψZ) − T c(X, ψY, Z) 2 2 2 +aη(X)F (Y, ψZ) + aη(Y )F (ψZ, X) + aη(X)F (ψY, Z) + aη(Z)F (X, ψY ) − ∇g 2 = ( X F )(Y,Z) + aη(Y )g(ψZ, ψX) + aη(Z)g(X, ψ Y ) − ∇g − = ( X F )(Y,Z) + aη(Y )g(Z,X) aη(Z)g(X,Y ). Thus we get the equation δF δF (∇g F )(Y,Z) = g(X,Z) (Y ) − g(X,Y ) (Z). X n − 1 n − 1 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 51 Since δF ◦ ψ = 0 we have δF = δF (ξ)η and obtain δF (ξ) δF (ξ) (∇g F )(Y,Z)η(Y ) − g(X,Y )( + 2a)η(Z) = (g(X,Z)η(Y ) − g(X,Y )η(Z)). X n − 1 n − 1 We summarize this result: An almost hermitian structure on M¯ , given by an almost contact structure on M is of class χ4 if and only if δF (ξ) δF (ξ) (∇g F )(Y,Z) = (g(X,Z)η(Y ) − g(X,Y )η(Z)), δF ◦ ψ = 0 and dη = 2 F. X n − 1 n − 1 The first condition implies the others: For some local orthonormal basis e1, . . . , en = ξ of TM we have ∑n ∑n g δF (ξ) δF (X) = − (∇ F )(ei,X) = − (g(ei, X)η(ei) − η(X)) ei n − 1 i=1 i=1 δF (ξ) = − (−nη(X) + η(X)) = δF (ξ)η(X) n − 1 ∇g δF (ξ) − ◦ and thus the condition ( X F )(Y,Z) = n−1 (g(X,Z)η(Y ) g(X,Y )η(Z)) implies δF ψ = 0. ∇g − ∇g ∇g Since ξ is a Killing vector field and thus ( X F )(ξ, ψY ) = F ( X ξ, ψY ) = g( X ξ, Y ) is skew symmetric in X and Y we have ∇g − ∇g ∇g − ∇g ∇g dη(X,Y ) = ( X η)(Y ) ( Y η)(X) = ( X F )(ξ, ψY ) ( Y F )(ξ, ψX) = 2( X F )(ξ, ψY ) ∇g δF (ξ) − and with condition ( X F )(Y,Z) = n−1 (g(X,Z)η(Y ) g(X,Y )η(Z)) we already get dη = δF (ξ) 2 n−1 F . Case χ1 ⊕ χ3: The condition for a structure of class χ1 ⊕χ3 can be obtained directly from Lemma 3.10. Case χ3 ⊕ χ4: An almost hermitian structure on M¯ is of class χ3 ⊕ χ4 if and only if N¯ = 0. Due to Lemma 3.7, this is equivalent to N = 0. Case χ1 ⊕ χ4: The condition for an almost hermitian structure to be of class χ1 ⊕ χ4 is the same as for the class χ4, setting X = Y : 1 1 T¯(X,JX,Y ) = [g¯(X,X)(δF (Y ) − a(n − 1)η(Y )) − g¯(X,Y )(δF (X ) − a(n − 1)η(X )) 2 n − 1 M M M M +g ¯(X,JY )(δF ((JX)M ) − a(n − 1)η((JX)M ))]. The equation is still linear in Y but not in X. We set X = V + b∂r for b ∈ R and V ∈ TM: 1 b 1 T¯(V,JV,Y ) + T¯(V, ξ, Y ) = [(b2 + a2r2g(V,V ))(δF (Y ) − a(n − 1)η(Y )) 2 2ar n − 1 M M 2 2 − (bg¯(∂r,Y ) + a r g(V,YM ))(δF (V ) − a(n − 1)η(V )) b b(n − 1) + (¯g(V,JY ) − barη(Y ))(−δF (ψV ) + δF (ξ) − )]. M ar r This is satisfied for any b if and only if 1 1 T¯(V,JV,Y ) = [a2r2g(V,V )(δF (Y ) − a(n − 1)η(Y )) 2 n − 1 M M 2 2 − a r g(V,YM )(δF (V ) − a(n − 1)η(V )) +g ¯(V,JY )(−δF (ψV ))] 52 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS and 1 1 T¯(V, ξ, Y ) = [−g¯(∂ , Y )(δF (V ) − a(n − 1)η(V )) 2ar n − 1 r (II.5) δF (ξ) (n − 1) +g ¯(V,JY )( − ) + arη(Y )δF (ψV )] ar r M and 2 0 = δF (YM ) − η(YM )δF (ξ) = δF (YM − η(YM )ξ) = −δF (ψ (YM )), where the last equation is satisfied if and only if δF ◦ ψ = 0. For Y ∈ TM with the condition δF ◦ ψ = 0 equation (II.5) leads to 1 δF (ξ) T (ξ,V,Y ) = F (V,Y )( − a). 2 n − 1 y − δF (ξ) Since ξ T = dη 2aF we have dη = 2 n−1 F and thus dF = 0. With Theorem 8.4 in [FI02] this implies N = 0 and the structure is already of class χ4. Thus a structure is never of class χ1 or of mixed class χ1 ⊕ χ4. We now compare the result of Theorem 3.11 with the 12 classes of almost contact structures given in Section 1 of Chapter I. We just consider manifolds admitting a characteristic connection (recall that Theorem 1.3 of Chapter I formulates the criterion for its existence). Theorem 3.12. If the almost hermitian structure on M¯ is • of class χ3, then the almost contact structure on M is of class C3 ⊕ .. ⊕ C8 but not of class C3 ⊕ C4 ⊕ C5 ⊕ C7 ⊕ C8 or of class C6. • of class χ1⊕χ3, then the almost contact structure on M is not of class C1⊕..⊕C5⊕C7⊕..⊕C12 nor of class C6. The almost hermitian structure on M¯ is • K¨ahlerif and only if the almost contact structure on M is α-Sasaki (of class C6) and δF (ξ) = a(n − 1). • of class χ4 if and only if the almost contact structure on M is an α-Sasaki structure. • of class χ3 ⊕ χ4 if and only if the almost contact structure on M is of class C3 ⊕ .. ⊕ C8 and there exists a characteristic connection. Furthermore the structure on M is Sasaki if and only if the almost hermitian structure on M¯ is of class χ4 with δω(ξ) = (a − 1)(n − 1). Proof. If the structure on M¯ is of class χ3, we have N = 0 and thus the structure on M is of class C3 ⊕ .. ⊕ C8. Furthermore, δF (ξ) = a(n − 1) holds, but on C3 ⊕ C4 ⊕ C5 ⊕ C7 ⊕ C8 we have δF (ξ) = 0 and a structure on M of class C6 implies a structure on M¯ of class χ4. A structure on M¯ of class χ1 ⊕ χ3 implies on M the relation δF (ξ) ≠ 0, but on C1 ⊕ .. ⊕ C5 ⊕ C7 ⊕ .. ⊕ C12 we have δF (ξ) = 0 and again a structure on M of class C6 implies a structure on M¯ of class χ4. ¯ ∇g − With Theorem 3.11, a structure on M is K¨ahlerianif and only if ( X F )(Y,Z) = aη(Y )g(X,Z) aη(Z)g(X,Y ) holds on M, which is equivalent for the almost contact structure to be of class C6 with δF (ξ) = a(n − 1). The condition of Theorem 3.11 for a structure of class χ4 on M is equivalent to the definition of an almost contact structure on M¯ to be of class C6. In C3 ⊕ .. ⊕ C8 we have N = 0, which together with the existence of a characteristic connection 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 53 is equivalent to the property that the structure on M¯ is of class χ3 ⊕ χ4. A structure on M is Sasaki if and only if it is of class C6 and δF (ξ) = n − 1. Due to Theorem 3.11 this is equivalent to the condition for the structure on M¯ to be of class χ4 with δω(ξ) = (a − 1)(n − 1). ¯ ∇g Remark 3.13. If we construct M with a = 1, we obtain a K¨ahlerianstructure, and ( X F )(Y,Z) = η(Y )g(X,Z) − η(Z)g(X,Y ) defines a Sasakian structure on M. This is the classical case treated by B¨arin [B¨a93]. 3.2 Corresponding spinors on metric almost contact structures and their cones We shall now work out in detail the abstract spinor correspondence stated in Lemma 2.4 for the case that M carries a metric almost contact structure. The following result serves as a preparation. Lemma 3.14. Given a metric almost contact structure with characteristic connection on M, the lift of η ∧ F to its cone M¯ is given by 1 (∂ yω) ∧ ω. a3r3 r y 1 y ∧ ∈ Proof. Since ∂r [ a3r3 (∂r ω) ω] = 0 we just need to show the equality on TM. For X,Y TM we have 1 1 F (X,Y ) = g(X, ψY ) = − g¯(X,JY + η(Y )ar∂ ) = − ω(X,Y ) a2r2 r a2r2 and 1 η(X) = g(X, arJ∂ ) = ω(X, ∂ ) r ar r − 1 − 1 y which proves F = a2r2 ω and η = ar ∂r ω on T M. We recall the definition of the connections ∇s ∇g c ∇¯ s ∇¯ g¯ ¯ X Y = X Y + 2sT (X,Y ) and X Y = X Y + 2sT (X,Y ) for s ∈ R from the beginning of this chapter. Theorem 3.2 yields T c = T + 2aη ∧ F and since T¯ = a2r2T and T c = a2r2T c, we get T c − T¯ as the lift of 2a3r2η ∧ F to M¯ . With Lemma 3.14 c − ¯ 2 y ∧ we obtain T T = r (∂r ω) ω. Theorem 3.15. Assume that the almost contact metric manifold (M, g, ψ, η) admits a charac- 1 − 1 teristic connection and is spin. Then there is for α = 2 a or α = 2 a: 1. A one to one correspondence between Killing spinors with torsion ∇s X ϕ = αXϕ ∇¯ s 4s y ∧ ¯ on M and parallel spinors of the connection + r (∂r ω) ω on M with cone constant a 2s ∇¯ s ϕ + (Xy(∂ yω) ∧ ω)ϕ = 0, X r r 2. A one to one correspondence between ∇¯ s-parallel spinors on M¯ with cone constant a and spinors on M satisfying ∇s − y ∧ X ϕ 2asX (η F )ϕ = αXϕ. 1 In particular, for s = 4 we get the correspondence 54 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS spinors on M spinors on M¯ ∇c ∇¯ − 1 y y ∧ X ϕ = αXϕ X ϕ = 2r X ((∂r ω) ω)ϕ ∇c a y ∧ ∇¯ X ϕ = αXϕ + 2 X (η F )ϕ X ϕ = 0 Remark 3.16. Assuming a Killing spinor with torsion on M, we get the correspondence to a spinor on M¯ as in case (1) of the Theorem. In particular this spinor is parallel in the cone direction. ∇¯ ∇¯ g¯ 1 ¯ Remark 3.17. Since = + 2 T is the characteristic connection of the almost hermitian structure on M¯ , we can write T¯ = N¯ + dωJ , where dωJ = dω ◦ J. Thus one can rewrite all equations above. For example the correspondence (1) of Theorem 3.15 is given with spinors on M¯ satisfying 2 ∇¯ g¯ ϕ + sXy[N¯ + dωJ + (∂ yω) ∧ ω]ϕ = 0. X r r Equivalently, one can use the description of T c on M given by T c = η ∧dη +dF ψ +N −η ∧(ξyN) ([FI02]) to rewrite the second correspondence. Note that this also implies that T¯ = N¯ + dωJ is the lift of a2r2T = a2r2(T c − 2aη ∧ F ) = a2r2(η ∧ (dη − 2aF ) + dF ψ + N − η ∧ (ξyN)) J to M¯ , in particular we have ∂ry(N¯ + dω ) = 0. 3.3 Examples In this Section, we shall discuss several examples of metric almost contact structures and the special spinor fields that exist on them and on their cones. In particular, we shall describe several situations where the cone carries a parallel spinor field for the characteristic connection ∇¯ of its almost hermitian structure. Example 3.18. For a metric almost contact manifold (M, g, ψ, η), the deformation 1 g := tg + (t2 − t)η ⊗ η, ξ := ξ, η := tη, t > 0 t t t t is often used for different purposes and constructions (compare Example 1.10). Since Tanno used it in [Ta68] it is the so called Tanno deformation. It has the property that if the original manifold is K-contact or Sasaki, then the deformed manifold (M, gt, ξt, ηt, ψ) has again this property. In [Be12, Cor.2.18] it was proved that almost any Sasakian η-Einstein manifold satisfying Ricg = λg − νη ⊗ η for some λ, ν ∈ R carries Killing spinors with torsion. On an Einstein-Sasaki manifold (M, g, ψ, η) of dimension n = 2k + 1 > 5 this spinors are constructed as follows. Consider the one dimensional subbundles of the spinor bundle Σt of (M, gt) defined by L1(Σt) := {ϕ ∈ Σt | ψ(X)ϕ = −iXϕ ∀X ⊥ ξ},L2(Σt) := {ϕ ∈ Σt | ψ(X)ϕ = iXϕ ∀X ⊥ ξ}. k+1 Define ϵ = ±1 to be the number satisfying e1ψ(e1)...ekψ(ek)ξϕ = ϵi ϕ for a local orthonormal frame e1, ψ(e1), .., ek, ψ(ek), ξ on M. Theorem 2.22 from [Be12] then states that the spinors 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 55 ∈ ∈ k+1 1 − ϕ1 L1(Σt) and ϕ2 L2(Σt) are Killing spinors with torsion for st = 4(k−1) ( t 1) with Killing numbers ϵ 2kt − (k + 1) ϵ β = = (1 − 4s ) and β = (−1)k+1β (∗) 1,t 2 t(k − 1) 2 t 2,t 1,t respectively. For t = 1, there is no deformation, and indeed the parameter st is then zero and the two spinors are just classical Riemannian Killing spinors. Since (M, gt, ξt, ηt, ψ) with fundamental c c 2-form Ft is Sasakian, the characteristic torsion of ∇ is given by T = ηt ∧ dηt = 2ηt ∧ Ft. Thus, the Killing equation ∇gt y c X ϕi + st(X T )ϕi = βi,tXϕi, i = 1, 2 can equivalently be reformulated as 1 1 ∇gt ϕ + (XyT c)ϕ − (1 − 4s ) (XyT c)ϕ = β Xϕ . X i 4 i t 4 i i,t i If 1 − 4st = 0, both Killing numbers βi,t vanish by equation (∗) and the Killing equation is c c reduced to ∇ ϕi = 0 – the spinor fields ϕi are ∇ -parallel and, as observed before, the cone − k+1 construction is not possible. The condition 1 4st > 0 is equivalent to t > 2k and we observe that in this case, the last equation is exactly of the form treated in Theorem 3.15, case (2) for s = 1/4 and a = 2|βi,t| = 1 − 4st > 0. Recall that we know from Theorem 3.12 that the cone (M,¯ g¯t) of the Tanno deformation is a locally conformally K¨ahlermanifold (class χ4). Hence, we can conclude from Theorem 3.15, case (2): Theorem 3.19. Let (M, g, ψ, η) be an Einstein Sasaki manifold of dimension 2k + 1 > 5. k+1 ¯ Consider its Tanno deformation (M, gt, ξt, ηt, ψ) for t > 2k and the cone (M, g¯t,Jt), constructed with cone constant a = 1−4st > 0, and endowed with the conformally K¨ahlerstructure described before. Then the two Killing spinors with torsion on (M, gt, ξt, ηt, ψ) induce each a spinor on the cone (M,¯ g¯t,Jt) that is parallel with respect to its characteristic connection ∇¯ . k+1 Although Killing spinors with torsion do exist on (M, gt, ξt, ηt, ψ) for 0 < t < 2k , Theorem 3.15, case (2) cannot be applied because the signs do not match. Of course, case (1) does still hold and therefore we obtain a spinor field satisfying a more complicated equation on M¯ . For t = 1 (meaning st = 0), Theorem 3.19 is the classical cone correspondence between Riemannian Killing spinors on Einstein-Sasaki manifolds and Riemannian parallel spinors on their cone [B¨a93]. Example 3.20. The Heisenberg group H is defined to be the following Lie subgroup of Gl(5, R): 1 u v z 0 1 0 x | ∈ R H := u, v, x, y, z . 0 0 1 y 0 0 0 1 The vector fields u1 = ∂u, u2 = ∂x + u∂z, u3 = ∂v, u4 = ∂y + v∂z and u5 = ∂z form a basis of the left invariant vector fields. For ρ > 0 we consider the metric ([KV85]) 1 g = (du2 + dx2 + dv2 + dy2) + (dz − udx − vdy)2 ρ √ √ √ √ and get an orthonormal frame e1 = ρu1, e2 = ρu2, e3 = ρu3, e4 = ρu4 and e5 = u5. We consider the almost contact structures given by F1 := e1 ∧ e2 − e3 ∧ e4 and F2 := e1 ∧ e2 + e3 ∧ e4, 56 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS both with the same ξ := e5. Becker-Bender calculates in [Be12] that the characteristic connection c for both structures is given by its torsion T = −ρe1 ∧ e2 ∧ e5 − ρe3 ∧ e4 ∧ e5. She also proves that ϕ1 and ϕ2, defined via the equations ψ2(X)ϕ1 = −iXϕ1 ∀X ⊥ ξ and ψ2(X)ϕ2 = iXϕ2 ∀X ⊥ ξ, where ψj is the (1, 1) tensor to the 2-form Fj for j = 1, 2, are Killing spinors with torsion for − 3 − s = 4 with Killing number ρ and ρ respectively: − 3 − 3 ∇ 4 ∇ 4 − X ϕ1 = ρXϕ1 and X ϕ2 = ρXϕ2 − 3 ∇ 4 ∇g − 3 c for := 2 T . On M¯ we get two almost hermitian structures, constructed from F1 and F2: We consider the ¯ 1 orthonormal basis on M given by Xi := ar ei for i = 1..5 and X6 := ∂r. Then the almost hermitian structures on M¯ are given by Ω1 = −X1 ∧ X2 + X3 ∧ X4 + X5 ∧ X6 and Ω2 = −X1 ∧ X2 − X3 ∧ X4 + X5 ∧ X6. We look at the corresponding characteristic connections ∇¯ 1 and ∇¯ 2, coming from the connections 1 2 1 c 2 c ∇ and ∇ with torsions T = T − 2aη ∧ F1 and T = T − 2aη ∧ F2 on M, and the s-dependent connections ∇¯ s,1 := ∇¯ g + 2sT¯1 and ∇¯ s,2 := ∇¯ g + 2sT¯2. The equivalence of the characteristic ∇¯ s,i 4s y ∧ connections for F1 and F2 on M implies, that the connections + r (∂r Ω) Ω are the same for i = 1, 2. With Theorem 3.15 we get for both orientations on M¯ , constructed with a = 2ρ, the existence of a spinor ϕ satisfying − 3 ,i 3 ∇¯ 4 ϕ − Xy((∂ yΩ ) ∧ Ω )ϕ = 0 for i = 1, 2. X 2r r i i Thus we have two linear independent spinors on M¯ satisfying this equation, which, due to Remark 3.17 is equivalent to 3 2 ∇¯ g¯ ϕ − Xy[N¯ + dΩ + (∂ yΩ ) ∧ Ω ]ϕ = 0 X 4 i i r r i i where N¯i denotes the Nijenhuis tensor of the almost hermitian structure Ωi. We calculate the types of the structures F1 and F2 and with Theorem 3.12 we get immediately Lemma 3.21. The structure F1 is of type C7 and the structure F2 is of type C6. Thus the almost hermitian structure on M¯ induced by F1 is of mixed type χ3 ⊕χ4 and the almost hermitian structure on M¯ induced by F2 is of type χ4. Proof. With the given Fi we get in the basis defined above ψie1 = −e2 for i = 1, 2 and ψ1e3 = e4, ψ2e3 = −e4; the other values of ψkej we get from the skew symmetry of ψk. With the equation ∇g 1 c 1 c c − ∧ ∧ − ∧ ( X F2)(Y,Z) = 2 T (X, ψ2Y,Z) + 2 T (X, Y, ψ2Z) and the fact that T = ρe1 e2 e5 ρe3 e4 ∧ e5 = −ρF2 ∧ η we get ∑ ∑ g 1 c δFk(ξ) = − (∇ Fk)(ei, ξ) = − T (ei, ψkei, ξ) ei 2 i i 1 ∑ 1 ∑ = ρ (e ∧ e ∧ e + e ∧ e ∧ e )(e , ψ e , ξ) = ρ (e ∧ e + e ∧ e )(e , ψ e ) 2 1 2 5 3 4 5 i k i 2 1 2 3 4 i k i i i and thus we have δF1(ξ) = 0 and δF2(ξ) = −2ρ. We calculate 1 1 (∇g F )(Y,Z) = T c(X, ψ Y,Z) + T c(X, Y, ψ Z) X k 2 k 2 k 1 = − ρ[F ∧ η(X, ψ Y,Z) + F ∧ η(X, Y, ψ Z)] 2 2 k 2 k 1 = − ρ[F (X, ψ Y )η(Z) + F (ψ Y,Z)η(X) + F (Y, ψ Z)η(X) + F (ψ Z,X)η(Y )]. 2 2 k 2 k 2 k 2 k 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 57 Since ψ1 and ψ2 commute we have F2(X, ψkY ) = g(X, ψ2ψkY ) = g(X, ψkψ2Y ) = −g(ψkX, ψ2Y ) = −F2(ψkX,Y ) and get 1 (∇g F )(Y,Z) = − ρ[F (X, ψ Y )η(Z) − F (X, ψ Z)η(Y )]. (II.6) X k 2 2 k 2 k The structure F1 is of type C7 if δF1(ξ) = 0 and (∇g F )(Y,Z) = η(Z)(∇g F )(X, ξ) − η(Y )(∇g F )(ψ Z, ξ). X 1 Y 1 ψ1X 1 1 The right side equals 1 1 [η(Z)T c(Y, ψ X, ξ) − η(Y )T c(ψ X, ψ2Z, ξ)] = − ρ[η(Z)F (Y, ψ X) + η(Y )F (ψ X,Z)] 2 1 1 1 2 2 1 2 1 and due to the calculation above, this proves the first statement. With the equation (II.6) we have 1 δF (ξ) (∇g F )(Y,Z) = − ρ(g(X, ψ2Y )η(Z)−g(X, ψ2Z)η(Y )) = − 2 (g(X,Y )η(Z)−g(X,Z)η(Y )) X 2 2 2 2 4 which proves that F2 is of type C6. Example 3.22. Another example (see [Be12]) is given by the homogeneous space M := SO(3)× SL(2, R)/SO(2) with the embedding [ ] ( )− cos t − sin t t 1 SO(2) ∋ A(t) := 7→ A(t),A . sin t cos t 2 As an orthonormal basis of a reductive complement of so(2) in so(3) × sl(2, R) we choose 0 0 0 0 0 1 1 1 0 − e1 := D1 0 0 1 , 0 , e2 := D1 0 0 0 , 0 , e3 := D2 0, , 2 0 −1 0 1 0 −1 0 0 − 0 1 0 1 0 1 1 0 −1 e4 := D2 0, , e5 := c1 1 0 0 , c2 , 2 1 0 2 1 0 0 0 0 ̸ 2 2 − − such that c1 + c2 = 0, D1 = c1(c1 + c2), D2 = c2(c1 + c2) and the numbers c1, c2 and (c1 + c2) have the same signature. We consider the almost contact structure (M, ξ, F ) defined via ξ := e5 and F = e1 ∧ e2 + e3 ∧ e4. c c Then the characteristic connection ∇ has torsion T = −c1e1 ∧ e2 ∧ e5 − c2e3 ∧ e4 ∧ e5. Lemma 3.23. The almost contact structure (M, ξ, F ) is normal. Furthermore, the almost her- − − ¯ c1 c2 mitian structure on M, constructed with a = 4 , induced by the almost contact structure (M, ξ, F ) is of class χ3 and thus the structure (M, ξ, F ) is of mixed class C3 ⊕ .. ⊕ C8. 58 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Proof. We use Theorem 3.11 and prove that the almost contact structure (M, ξ, F ) is normal, − − ∇g − satisfies δF = ( c1 c2)η and never satisfies ( X F )(Y,Z) = aη(Y )g(X,Z) aη(Z)g(X,Y ) − − ¯ c1 c2 (Thus the structure on M is never K¨ahlerand for a = 4 it really is of type χ3). First, the definition of F implies ψ(e1) = −e2, ψ(e2) = e1, ψ(e3) = −e4, ψ(e4) = e3, ψ(e5) = 0 and thus e1(ψ(X)) = e2(X), e2(ψ(X)) = −e1(X), e3(ψ(X)) = e4(X), e4(ψ(X)) = −e3(X) and we get c T (X, ψY, ψZ) =η(X)(−c1e1 ∧ e2(ψY, ψZ) − c2e3 ∧ e4(ψY, ψZ)) =η(X)(−c1e1 ∧ e2(Y,Z) − c2e3 ∧ e4(Y,Z)). This implies c c T (X,Y,Z) = (−c1e1 ∧ e2 ∧ η − c2e3 ∧ e4 ∧ η)(X,Y,Z) = SX,Y,Z T (X, ψY, ψZ), where S denotes the cyclic sum. We have ∇g − ∇g ∇g − ∇g N(X,Y,Z) =( X F )(Z, ψY ) ( Y F )(Z, ψX) + ( ψX F )(Z,Y ) ( ψY F )(Z,X) ∇g − ∇g + η(X)( Y F )(ξ, ψZ) η(Y )( X F )(ξ, ψZ). ∇g 1 c c Using the equality ( X F )(Y,Z) = 2 (T (X, ψY, Z) + T (X, Y, ψZ)) we get c c N(X,Y,Z) = T (X,Y,Z) − SX,Y,Z T (X, ψY, ψZ) = 0. Secondly we calculate ∑ ∑ g 1 c δF (X) = − (∇ F )(ei,X) = − T (ei, ψei,X) ei 2 i i 1 = − [−c e ∧ e ∧ e (e , −e , X) − c e ∧ e ∧ e (e , e ,X) 2 1 1 2 5 1 2 1 1 2 5 2 1 − c2e3 ∧ e4 ∧ e5(e3, −e4,X) − c2e3 ∧ e4 ∧ e5(e4, e3,X) =η(X)(−c1 − c2)]. ∇g − Finally, to show that (M, ξ, F ) never satisfies ( X F )(Y,Z) = aη(Y )g(X,Z) aη(Z)g(X,Y ) we calculate g c1 (∇ F )(e1, ξ) = and aη(e1)g(e1, ξ) − aη(ξ)g(e1, e1) = −a e1 2 as well as g c2 (∇ F )(e3, ξ) = and aη(e3)g(e3, ξ) − aη(ξ)g(e3, e3) = −a. e3 2 Thus we can choose a such that the structure on M¯ is integrable if c1 = c2. But in the construc- tion of M we needed the condition that c1 and −c2 have the same signature and that c1 +c2 ≠ 0, which contradicts c1 = c2. ∇s In this example we only have Killing spinors with torsion satisfying X ϕ = αXϕ for α = 0. But since the construction of M¯ explicitly depends on 2α = a ≠ 0, we cannot lift these spinors to M¯ . 3. ALMOST HERMITIAN CONES OVER ALMOST CONTACT MANIFOLDS 59 3.4 Metric almost contact 3-structures Let M be a manifold of dimension n = 4m − 1 with 3 metric almost contact structures given by ξi, ηi and ψi for i = 1, 2, 3 (see [Ca09] for a more detailed description of metric almost contact 3-structures). Looking at the cone M¯ , we define the three almost hermitian structures J1(ar∂r) := ξ1,J1(ξ1) = −ar∂r,J1(V ) = −ψ1(V ) for V ⊥ ξ1, ∂r, J2(ar∂r) := ξ2,J2(ξ2) = −ar∂r,J2(V ) = −ψ2(V ) for V ⊥ ξ2, ∂r, J3(ar∂r) := −ξ3,J3(ξ3) = ar∂r,J3(V ) = −ψ3(V ) for V ⊥ ξ3, ∂r. Conversely, let M¯ be a 4m dimensional manifold with three almost hermitian structures J1, J2 and J3. We can define three almost contact structures ξ1 := +aJ1(∂r), ψ1(X) := −J1(X) +g ¯(J1(X), ∂r)∂r. ξ2 := +aJ2(∂r), ψ2(X) := −J2(X) +g ¯(J2(X), ∂r)∂r. ξ3 := −aJ3(∂r), ψ3(X) := +J3(X) − g¯(J3(X), ∂r)∂r. on M = M × {1} ⊂ M¯ . A 4m-dimensional manifold with three almost hermitian structures ωi =g ¯(., Ji.) is called hyper- K¨ahlerwith torsion (HKT) if the almost hermitian structures are integrable (N¯i = 0) and J1 ◦ dω1 = J2 ◦ dω2 = J3 ◦ dω3. We can apply Theorem 3.2 to each of these structures and prove Theorem 3.24. The three almost hermitian structures on M¯ satisfy the relation J1J2 = −J2J1 = J3 if and only if ξ1, ξ2 and ξ3 are orthonormal and the almost contact structures on M satisfy the following ψ3ψ2 = −ψ1 + η2 ⊗ ξ3, ψ2ψ3 = +ψ1 + η3 ⊗ ξ2, ψ1ψ3 = −ψ2 + η3 ⊗ ξ1, (II.7) ψ3ψ1 = +ψ2 + η1 ⊗ ξ3, ψ2ψ1 = −ψ3 + η1 ⊗ ξ2, ψ1ψ2 = +ψ3 + η2 ⊗ ξ1, (II.8) where ηi is the dual to ξi for i = 1, 2, 3. The appendant connection ∇¯ satisfies ∇¯ J2 = ∇¯ J3 = c,i ∇¯ J1 = 0 if and only if the characteristic connections ∇ on M of the three almost hermitian i structures (ηi, ψi) are such that the corresponding connections ∇ constructed in Definition 3.1 coincide ∇1 = ∇2 = ∇3 =: ∇. In this case, we get the additional commutator relations [ξ1, ξ2] = 2aξ3 − T (ξ1, ξ2), [ξ2, ξ3] = 2aξ1 − T (ξ2, ξ3), [ξ3, ξ1] = 2aξ2 − T (ξ3, ξ1). If furthermore the almost contact structures are normal, the three almost hermitian structures on M¯ form an HKT structure. Proof. Given three almost hermitian structures satisfying the relation J1J2 = −J2J1 = J3, we compute ψ3(ψ2(X)) = −J3(J2(X)) +g ¯(J3(J2(X)), ∂r)∂r +g ¯(J2(X), ∂r)J3(∂r) − g¯(J3(X), ∂r)¯g(J3(∂r), ∂r)∂r 2 = −ψ1(X) − g¯(X,J2∂r)J3(∂r) = −ψ1(X) − a g(X,J2∂r)J3(∂r) = −ψ1(X) + g(X, ξ2)ξ3, and similarly for the other relations. Conversely, given three almost hermitian structures satis- fying equations (II.7) and (II.8) we plug in ξ1, ξ2, and ξ3 and, with ψi(ξi) = 0 for i = 1, 2, 3, we obtain immediately ψ1(ξ2) = ξ3, ψ1(ξ3) = −ξ2, ψ2(ξ1) = −ξ3, ψ2(ξ3) = ξ1, ψ3(ξ1) = ξ2, ψ3(ξ2) = −ξ1. 60 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Since all ψi leave the vector space V := span(ξ1, ξ2, ξ3) invariant and since they are orthonormal, ⊥ they also leave V invariant. For X ⊥ ξ1, ξ2, ξ3, ∂r we have J1(J2(X)) = ψ1(ψ2(X)) = ψ3(X) = J3(X) = −ψ2(ψ1(X)) = −J2(J1(X)). For ξ1 we obtain J1(J2(ξ1)) = −J1(ψ2(ξ1)) = J1(ξ3) = −ψ1(ξ3) = ξ2 = J2(ar∂r) = −J2(J1(ξ1)) = ψ3(ξ1) = J3(ξ1) and similarly for ξ2, ξ3 and ∂r. For a connection as in Theorem (3.2), we have that all almost hermitian structures are parallel under ∇¯ and for X,Y ∈ TM ∇¯ g¯ − ∇¯ g¯ ∇¯ − ∇¯ − ¯ [X,Y ] = X Y Y X = X Y Y X T (X,Y ). Thus the commutator relations are given by 2 2 ∇¯ − ∇¯ − ¯ [ξ1, ξ2] = a [J1(∂r),J2(∂r)] = a ( J1(∂r )J2(∂r) J2(∂r )J1(∂r)) T (ξ1, ξ2) 2 ∇¯ − ∇¯ − ¯ = a (J2( J1(∂r )∂r) J1( J2(∂r )∂r)) T (ξ1, ξ2) 2 = a (J2(J1(∂r)) − J1(J2(∂r))) − T¯(ξ1, ξ2) 2 = −2a J3(∂r) − T¯(ξ1, ξ2) = 2aξ3 − T (ξ1, ξ2). The other relations are to be calculated similarly. If the almost contact structures are normal, then the almost hermitian structures are normal and with the formula for the torsion given in Remark 3.17 we have 1 g¯(∇¯ ,Y,Z) =g ¯(∇¯ g¯ Y,Z) + (J ◦ dω )(X,Y,Z) X X 2 i i for any i = 1, 2, 3. This implies J1 ◦ dω1 = J2 ◦ dω2 = J3 ◦ dω3. Remarks 3.25. - In [FFUV11, Section 7], the authors obtain a similar result (but without a description of the characteristic connections). Furthermore, they investigate more closely the conditions for the HKT structure to be strong (Ji ◦ dωi is closed). - If we rescale the metric such that a = 1 and if T = 0, we have 3 K¨ahlerianstructures on M¯ and thus 3 Sasakian structures on M. Then the commutator relations in Theorem 3.24 ensure that the structures on M form a 3-Sasakian structure. This is Lemma 5 of [B¨a93]:A one to one correspondence between hyper-K¨ahlerstructures on M¯ and 3-Sasaki structures on M. - We emphasize that it is not necessary that the three characteristic connections ∇c,i, i = 1, 2, 3 coincide in order to apply Theorem 3.24, only the connections ∇i with torsion T i = c,i i T − 2aηi ∧ Fi have to be equal. If M is a 3-Sasakian manifold, T = 0 for i = 1, 2, 3 and 1 2 3 g thus ∇ = ∇ = ∇ = ∇ . In this case there exists a special G2 structure on M which will be discussed in Example 4.18. 4 G2 structures – Spin(7) structures on the cone Let (M, g, Ψ,P ) be a G2 manifold (see Section 4 of Chapter I). For Spin(7) structures there is no one to one correspondence to spinors. Since we want to lift a G2 structure to a Spin(7) structure on the cone, in this section it is more convenient to use the description of a G2 structures via the differential form Ψ rather then via a spinor. Correspondences of geometric structures on non-twisted cones using the differential forms are considered in many other cases, see [II05] to name but one. We cite a classical, but for us crucial result by Fernandez and Gray: 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 61 Lemma 4.1 ([FG82, Lemma 2.7]). ∗Ψ(V, W, X, Y ) = g(P (V,W ),P (X,Y )) − g(V,X)g(W, Y ) + g(V,Y )g(W, X) = Ψ(V, W, P (X,Y )) − g(V,X)g(W, Y ) + g(V,Y )g(W, X). Remark 4.2. In [FG82] this formula is stated differently, ∗Ψ(V, W, X, Y ) = −g(P (V,W ),P (X,Y )) + g(V,X)g(W, Y ) − g(V,Y )g(W, X). = −Ψ(V, W, P (X,Y )) + g(V,X)g(W, Y ) − g(V,Y )g(W, X). This is due to the standard 3-form Ψ used by Fern´andezand Gray, which corresponds to the orientation opposite to ours. This changes the sign of the Hodge operator. Now we are able to prove Lemma 4.3. For any metric connection ∇ with skew torsion on M, the G2 form Ψ satisfies (∇Z ∗ Ψ)(V, W, X, Y ) = (∇Z Ψ)(V, W, P (X,Y )) + (∇Z Ψ)(X,Y,P (V,W )). If ∇ satisfies ∇Ψ = a ∗ Ψ for some a > 0, we have the simplified relation (∇Z ∗ Ψ)(V, W, X, Y ) = a[Ψ(X,Y,V )g(Z,W ) − Ψ(X,Y,W )g(Z,V ) +Ψ(V, W, X)g(Z,Y ) − Ψ(V, W, Y )g(Z,X)]. Proof. For any metric connection with skew torsion we have (∇Z ∗ Ψ)(V, W, X, Y ) =Z ∗ Ψ(V, W, X, Y ) − ∗Ψ(∇Z V, W, X, Y ) − ∗Ψ(V, ∇Z W, X, Y ) − ∗Ψ(V, W, ∇Z X,Y ) − ∗Ψ(V, W, X, ∇Z Y ). Since ∇ is metric, g is parallel and with Lemma 4.1 we get =ZΨ(V, W, P (X,Y )) − Ψ(∇Z V, W, P (X,Y )) − Ψ(V, ∇Z W, P (X,Y )) − Ψ(V, W, P (∇Z X,Y )) − Ψ(V, W, P (X, ∇Z Y )) − Ψ(V, W, ∇Z P (X,Y )) + Ψ(V, W, ∇Z P (X,Y )). We have Ψ(V, W, (∇Z P )(X,Y )) = g(P (V,W ), (∇Z P )(X,Y )) = (∇Z Ψ)(X,Y,P (V,W )) and thus we get (∇Z ∗ Ψ)(V, W, X, Y ) = (∇Z Ψ)(V, W, P (X,Y )) + (∇Z Ψ)(X,Y,P (V,W )). The condition ∇Ψ = a ∗ Ψ implies (∇Z ∗ Ψ)(V, W, X, Y ) = −a ∗ Ψ(P (X,Y ),Z,V,W ) − a ∗ Ψ(P (V,W ),Z,X,Y ) and aplying once again Lemma 4.1 yields (∇Z ∗ Ψ)(V, W, X, Y ) = = −aΨ(P (X,Y ),Z,P (V,W )) − aΨ(P (V,W ),Z,P (X,Y )) + ag(P (X,Y ),V )g(Z,W ) −ag(P (X,Y ),W )g(Z,V ) + ag(P (V,W ),X)g(Z,Y ) − ag(P (V,W ),Y )g(Z,X) = a[Ψ(X,Y,V )g(Z,W ) − Ψ(X,Y,W )g(Z,V ) + Ψ(V, W, X)g(Z,Y ) − Ψ(V, W, Y )g(Z,X)], which finishes the proof. We define a 4-form on the cone M¯ via 3 3 4 4 Φ(∂r,X,Y,Z) := a r Ψ(X,Y,Z), Φ(X,Y,Z,W ) := a r ∗ Ψ(X,Y,Z,W ) 62 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS ∈ y ⊥ for X,Y,Z,W TM. Since ∂r Φ locally is a G2-structure on ∂r , Φ is a Spin(7)-structure on M¯ . As in Section 3, given a characteristic connection on M with respect to Ψ, we construct a connection ∇ with skew symmetric torsion T on M such that its lift ∇¯ to M¯ with torsion T¯ is the ¯ ¯ ¯ characteristic connection on M with respect to Φ. Since we have T = T|TM and ∂ryT = 0, we have T¯ = T = 0 in case of a parallel Spin(7) structure with respect to the Levi-Civita connection on M¯ , and thus ∇ is the Levi-Civita connection on M and thus T measures the difference of the G2 structure to the nearly parallel case (see Remark 4.8). c Definition 4.4. Let (M, g, Ψ) be a G2 manifold with characteristic connection ∇ . We define a metric connection ∇ with skew symmetric torsion T via 2a T := T c − Ψ. 3 As in the metric almost contact case (see the comments in Definition 3.1), T cannot be computed abstractly, but it is found through an educated guess and justified a posteriori from its properties. Theorem 4.5. The connection ∇ satisfies ∇Ψ = a ∗ Ψ, and Φ is parallel with respect to ∇¯ , the appendant connection on M¯ . Proof. We have for the Riemannian connection ∇g on M ∇ − ∇g − ∇g − ∇g X Ψ(Y,Z,W ) = XΨ(Y,Z,W ) Ψ( X Y,Z,W ) Ψ(Y, X Z,W ) Ψ(Y,Z, X W ) 1 1 1 − Ψ(T (X,Y ),Z,W ) − Ψ(Y,T (X,Z),W ) − Ψ(Y,Z,T (X,W )) 2 2 2 1 = (∇c Ψ)(Y,Z,W ) + Ψ((T c − T )(X,Y ),Z,W ) X 2 1 1 + Ψ(Y, (T c − T )(X,Z),W ) + Ψ(Y,Z, (T c − T )(X,W )) 2 2 and because ∇cΨ = 0 we have ∇X Ψ(Y,Z,W ) = 1 = [(T c − T )(X,Y,P (Z,W )) + (T c − T )(X,Z,P (W, Y )) + (T c − T )(X, W, P (Y,Z))] 2 a = [Ψ(X,Y,P (Z,W )) + Ψ(X,Z,P (W, Y )) + Ψ(X, W, P (Y,Z))]. 3 With Lemma 4.1 we obtain a a ∗ Ψ(X,Y,Z,W ) = [∗Ψ(X,Y,Z,W ) + ∗Ψ(X, Z, W, Y ) + ∗Ψ(X, W, Y, Z)] 3 a = [Ψ(X,Y,P (Z,W )) + Ψ(X,Z,P (W, Y )) + Ψ(X, W, P (Y,Z)) 3 −g(X,Z)g(Y,W ) + g(X,W )g(Y,Z) − g(X,W )g(Z,Y ) + g(X,Y )g(Z,W ) −g(X,Y )g(W, Z) + g(X,Z)g(W, Y )] = ∇X Ψ(Y,Z,W ), which proves the first statement. To show ∇¯ Φ = 0 on M¯ we look at several cases. Let always be V, W, X, Y, Z ∈ TM. Case 1: If ∂r is one of the arguments, we compute 1 (∇¯ Φ)(∂ ,X,Y,Z) = W a3r3Ψ(X,Y,Z) − Φ(W,X,Y,Z) − r3a3Ψ(∇ X,Y,Z) W r r W 3 3 3 3 − r a Ψ(X, ∇W Y,Z) − r a Ψ(X,Y, ∇W Z) 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 63 1 1 = a3r3(∇ Ψ)(X,Y,Z) − Φ(W,X,Y,Z) = a4r3 ∗ Ψ(W, X, Y, Z) − Φ(W,X,Y,Z) = 0. W r r Case 2: If the direction of the derivative is equal to ∂r, we obtain 1 (∇¯ Φ)(X,Y,Z,W ) = ∂ (a4r4 ∗ Ψ(X,Y,Z,W )) − 4 Φ(X,Y,Z,W ) ∂r r r 1 = 4r3a4 ∗ Ψ(X,Y,Z,W ) − 4 Φ(X,Y,Z,W ) = 0. r Case 3: If the direction of the derivative and one argument are equal to ∂r we compute 1 (∇¯ Φ)(∂ ,X,Y,Z) = ∂ (a3r3Ψ(X,Y,Z)) − 3a3r3 Ψ(X,Y,Z) = 0. ∂r r r r Case 4: On T M we have: (∇¯ V Φ)(W, X, Y, Z) = 4 4 = a r V ∗ Ψ(W, X, Y, Z) − Φ(∇¯ V W, X, Y, Z) − Φ(W, ∇¯ V X,Y,Z) − Φ(W, X, ∇¯ V Y,Z) −Φ(W, X, Y, ∇¯ V Z) 1 1 = a4r4V ∗ Ψ(W, X, Y, Z) − Φ(∇ W − g¯(V,W )∂ ,X,Y,Z) − Φ(W, ∇ X − g¯(V,X)∂ ,Y,Z) V r r V r r 1 1 −Φ(W, X, ∇ Y − g¯(V,Y )∂ ,Z) − Φ(W, X, Y, ∇ Z − g¯(V,Z)∂ ) V r r V r r 4 4 4 5 = a r (∇V ∗ Ψ)(W, X, Y, Z) + r a [g(V,W )Ψ(X,Y,Z) − g(V,X)Ψ(W, Y, Z) +g(V,Y )Ψ(W, X, Z) − g(V,Z)Ψ(W, X, Y )], which is equal to zero due to Lemma 4.3. Conversely, given a Spin(7) structure (M,¯ g,¯ Φ, P,¯ p¯) on M¯ (see Section 5 of Chapter I for the 2 definitions), ∂ryΦ is a G2 structure with respect to the metric a g on M = M × {1} ⊂ M¯ and thus 1 Ψ := ∂ yΦ a3 r defines a G2 structure on M with respect to the metric g. To prove the following theorem, we need Lemma 4.6. If ∗ is the Hodge operator on M with respect to g and ∗a2g is the Hodge operator on M with respect to the metric a2g, we have for any 3-form ω ∗a2gω = a ∗ ω. i Proof. Let ei for i = 1..7 be an orthonormal basis with dual basis e on M with respect to g. 1 i 2 {i,j,k} i ∧ j ∧ k Then a ei with dual ae is a orthonormal basis with respect to a g. We define e := e e e and e{i,j,k,j} := ei ∧ ej ∧ ek ∧ el as well as (se){i,j,k} := sei ∧ sej ∧ sek for s ∈ R and (se){i,j,k,j} respectively. Then we have {i,j,k} 1 {i,j,k} 1 {1,..,7}\{i,j,k} 1 4 {1,..,7}\{i,j,k} {i,j,k} ∗ 2 e = ∗ 2 (ae) = (ae) = a e = a ∗ e , a g a3 a g a3 a3 which proves the lemma. Theorem 4.7. Given a Spin(7) structure on M¯ with characteristic connection ∇¯ being the lift of a connection ∇ on M, we have for the G2 structure Ψ induced by Φ ∇Ψ = a ∗ Ψ c 2a and the characteristic connection on (M, g, Ψ) is given by T = T + 3 Ψ. 64 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS Proof. We have for W, X, Y, Z ∈ TM 1 (∇ Ψ)(X,Y,Z) = [W Φ(∂ , X,Y,Z) W a3 r − Φ(∂r, ∇W X,Y,Z) − Φ(∂r,X, ∇W Y,Z) − Φ(∂r,X,Y, ∇W Z)] 1 1 = [(∇¯ Φ)(∂ , X,Y,Z) + Φ(∇¯ ∂ ,X,Y,Z)] = Φ(W,X,Y,Z). a3 W r W r a3 3 With Lemma 8 of [B¨a93]and the definition of Ψ we conclude Φ|TM = ∗a2g(∂ryΦ) = ∗a2g(a Ψ) = 4 ¯ 2 a ∗ Ψ, where ∗a2g is the Hodge operator on M ⊂ M with respect to the metric a g. The last equality follows from Lemma 4.6. Thus we get ∇Ψ = a ∗ Ψ. ∇c c 2a For the connection with torsion T = T + 3 Ψ we calculate as in the proof of Theorem 4.5 1 (∇c Ψ)(Y,Z,W ) = (∇ Ψ)(Y,Z,W ) + [(T − T c)(X,Y,P (Z,W )) + (T − T c)(X,Z,P (W, Y )) X X 2 + (T − T c)(X, W, P (Y,Z))] a = a ∗ Ψ(X,Y,Z,W ) − [Ψ(X,Y,P (Z,W )) 3 + Ψ(X,Z,P (W, Y )) + Ψ(X, W, P (Y,Z))] which is equal to zero due to Lemma 4.1. Since the characteristic connection of a G2 manifold is unique, this proves the theorem. c 2a Remark 4.8. In the case of an nearly parallel G2 structure we have T = 3 Ψ, i. e. T = 0 and thus ∇ = ∇g lifts to the Levi-Civita connection on M¯ and the corresponding Spin(7) structure c − 2a on the cone is integrable. This means, that, as in the metric almost contact case, T = T 3 Ψ measures the ’distance’ of the G2 structure from a nearly parallel G2 structure. 4.1 The classification of G2 structures and the corresponding classifi- cation of Spin(7) structures on the cone We will now discuss the classification of Fern´andez[Fe86] of Spin(7) structures on M¯ given in Section 5 of Chapter I, and compute the correspondence to the classification of G2 structures [FG82]. Again we are only interested in structures carrying a characteristic connection (G2 structures of class W1 ⊕ W3 ⊕ W4). We write XM for the projection on TM of a vector field X in T M¯ . We summarize some useful identities: Lemma 4.9. ∑ 1. P can be expressed through Ψ on TM: P (Y,Z) = l Ψ(el,Y,Z)el. 2. For any metric connection ∇˜ with skew torsion on M, we have: (∇˜ Ψ)(Y,Z,V ) = g((∇˜ P )(Y,Z),V ), X ∑ X ∑ (∇˜ X P )(Y,Z) = g(el, (∇˜ X P )(Y,Z))el = (∇˜ X Ψ)(el,Y,Z)el. l l ∑ ∇ ∇ ∗ 3. For , this can be simplified to ( X P )(Y,Z) = a l Ψ(X, el,Y,Z)el. 4. P, Ψ, and P¯ are related by (X,Y,Z ∈ TM) 3 3 P¯(∂r,X,Y ) ⊥ ∂r, g¯(P¯(X,Y,Z), ∂r) = Φ(X, Y, Z, ∂r) = −a r Ψ(X,Y,Z), 2 P¯(∂r,X,Y ) = arP (X,Y ), P¯(Y,Z,V )M = ar (∇Y P )(Z,V ). 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 65 5. The derivative of Φ on M¯ can be expressed in terms of Ψ on M (X,Y,Z,V,W ∈ TM): ∇¯ g¯ 3 3 ∇g − ∇ ( X Φ)(∂r,Z,V,W ) = a r [( )X Ψ](Z,V,W ), ∇¯ g¯ 4 4 ∇g − ∇ ∗ ( X Φ)(Y,Z,V,W ) = a r [( )X Ψ](Y,Z,V,W ). Proof. Statements (1)-(3) are easily checked. To prove statement (4) for X,Y,Z ∈ TM, we have 3 3 g¯(P¯(∂r,X,Y ),Z) = Φ(∂r,X,Y,Z) = a r Ψ(X,Y,Z) = arg¯(P (X,Y ),Z), thus P¯(∂r,X,Y ) = arP (X,Y ). Furthermore, 3 4 3 4 g¯(X, P¯(Y,Z,V )) = Φ(Y,Z,V,X) = a r (∇Y Ψ)(Z,V,X) = a r g(X, (∇Y P )(Z,V )) 2 = ar g¯(X, (∇Y P )(Z,V )), 2 and thus P¯(Y,Y,V )M = ar (∇Y P )(Z,V ). For (5) and vector fields X,Y,Z,V,W ∈ TM, we calculate ∇¯ g¯ 2( X Φ)(∂r,Z,V,W ) = = 2(∇¯ X Φ)(∂r,Z,V,W ) + Φ(∂r, T¯(X,Z),V,W ) + Φ(∂r,Z, T¯(X,V ),W ) + Φ(∂r,Z,V, T¯(X,W )) = a3r3[Ψ(T (X,Z),V,W ) + Ψ(Z,T (X,V ),W ) + Ψ(Z,V,T (X,W ))] 3 3 ∇ − ∇g ∇ − ∇g ∇ − ∇g = 2a r [Ψ(( X X )Z,V,W ) + Ψ(Z, ( X X )V,W ) + Ψ(Z,V, ( X X )W )] 3 3 g = −2a r [(∇ − ∇ )X Ψ](Z,V,W ), and similarly ∇¯ g¯ ( X Φ)(Y,Z,V,W ) = 1 = [Φ(T¯(X,Y ),Z,V,W ) + Φ(Y, T¯(X,Z),V,W ) + Φ(Y,Z, T¯(X,V ),W ) + Φ(Y,Z,V, T¯(X,W ))] 2 a4r4 = [∗Ψ(T (X,Y ),Z,V,W ) + ∗Ψ(Y,T (X,Z),V,W ) 2 + ∗ Ψ(Y,Z,T (X,V ),W ) + ∗Ψ(Y,Z,V,T (X,W ))] 4 4 g 4 4 g = −a r [(∇ − ∇ )X ∗ Ψ](Y,Z,V,W ) = a r [(∇ − ∇)X ∗ Ψ](Y,Z,V,W ), which finishes the proof. Remark 4.10. Since the characteristic connection of the Spin(7) structure on M¯ is unique (see Section 5 of Chapter I), we can conclude for any such structure satisfying ∇¯ g¯Φ = 0 that ∇ = ∇g g and thus ∇ Ψ = a ∗ Ψ and the G2 structure is of class W1. Conversely, given a connection ∇ ∇ ∗ ∇c c − 2a with skew symmetric torsion and Ψ = a Ψ we construct via T := T 3 Ψ, which satisfies ∇cΨ = 0 and thus is unique. Hence a metric connection with skew symmetric torsion and the property ∇Ψ = ∗Ψ is unique. For any tensor R on M we introduce the notation RxX to denote R(−,X). We extend the metric g to arbitrary k-tensors R,S via an orthonormal frame e1, . . . , en ∑n g(R,S) := R(ei1 , .., eik )S(ei1 , .., eik ). i1,..,ik=1 Lemma 4.11. A Spin(7) structure on M¯ is of class U1 if and only if on M • g(∇gΨ, ∗Ψ) = ag(∗Ψ, ∗Ψ), and • for every X ∈ TM we have g(∗Ψ, [(∇ − ∇g) ∗ Ψ]xX) = 3g(Ψ, [(∇ − ∇g)Ψ]xX). 66 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS The structure on M¯ is of class U2 if and only if the following conditions are satisfied for X,Y,Z,X1, .., X4 ∈ TM and a local orthonormal frame e1, .., e7 of TM: ∑7 • | ∇g − ∇ ∗ δΦ TM = 0 on TM, which is equivalent to 0 = [( )ei Ψ](ei,X,Y,Z) i=1 ∑4 ∑ i • 0 = (−1) δΨ(el, ej)Ψ(el, ej,Xi)Ψ(X1, .., Xˆi, .., X4) i=1 l g • 28[(∇ − ∇)W ∗ Ψ](X1,X2,X3,X4) ∑4 ∑ i+1 = (−1) δΨ(el, ej)Ψ(el, ej,Xi) ∗ Ψ(W, X1, .., Xˆi, .., X4). i=1 l 1 1 ¯ Proof. We consider a localg ¯-orthonormal framee ¯1 = ar e1, .., e¯7 = ar e7, e8 = ∂r of T M such that e1, .., e7 is a local orthonormal frame of TM. With Lemma 4.2 of [Fe86] a Spin(7) structure is defined to be of class U1 if and only if ∑8 − ∇¯ g¯ ¯ 0 = 6δΦ(¯p(X)) = ( e¯i Φ)(¯ej, e¯k, P (¯ei, e¯j, e¯k),X). i,k,j=1 For X ∈ TM we have ∑8 − ∇¯ g¯ ¯ 0 = 6δΦ(¯p(X)) = ( e¯i Φ)(¯ej, e¯k, P (¯ei, e¯j, e¯k),X) i,k,j=1 ∑7 ∑7 ∇¯ g¯ ¯ ∇¯ g¯ ¯ = ( e¯i Φ)(¯ej, e¯k, P (¯ei, e¯j, e¯k),X) + 2 ( e¯i Φ)(¯ej, ∂r, P (¯ei, e¯j, ∂r),X) i,k,j=1 i,j=1 ∑7 1 g¯ 2 = (∇¯ Φ)(ej, ek, ar (∇e P )(ej, ek) +g ¯(P¯(ei, ej, ek), ∂r)∂r,X) a6r6 ei i i,k,j=1 ∑7 1 g¯ +2 (∇¯ Φ)(ej, ∂r, arP (ei, ej),X) a4r4 ei i,j=1 1 ∑7 = a4r4[(∇g − ∇) ∗ Ψ](e , e , (∇ P )(e , e ),X) a5r4 ei j k ei j k i,k,j=1 ∑7 3 3 ∑7 1 g¯ a r g − Ψ(ei, ej, ek)(∇¯ Φ)(ej, ek, ∂r,X) − 2 ([∇ − ∇]e Ψ)(ej,P (ei, ej),X) a3r3 ei a3r3 i i,k,j=1 i,j=1 ∑7 ∇g − ∇ ∗ ∗ = [( )ei Ψ](ej, ek, Ψ(ei, el, ej, ek)el,X) i,k,j,l=1 ∑7 − ∇g − ∇ 3 Ψ(ei, ej, ek)([ ]ei Ψ)(ej, ek,X) i,k,j=1 = g(∗Ψ, (∇g − ∇) ∗ ΨxX) − 3g(Ψ, (∇g − ∇)ΨxX). 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 67 In case X = ∂r, we deduce from Lemma 4.9: ∑7 ∑7 0 = (∇¯ g¯ Φ)(e , e , P¯(e , e , e ), ∂ ) = ar2 (∇¯ g¯ Φ)(e , e , (∇ P )(e , e ), ∂ ) ei j k i j k r ei j k ei j k r i,j,k=1 i,j,k=1 ∑7 4 5 ∇g − ∇ ∇ =a r [( )ei Ψ](ej, ek, el)g(( ei P )(ej, ek), el) i,j,k,l=1 ∑7 ∑7 = − a4r5[ (∇g Ψ)(e , e , e )(∇ Ψ)(e , e , e ) − (∇ Ψ)(e , e , e )(∇ Ψ)(e , e , e )] ei j k l ei j k l ei j k l ei j k l i,j,k,l=1 i,j,k,l=1 = − a4r5[g(∇gΨ, ∇Ψ) − g(∇Ψ, ∇Ψ)] = −a5r5[g(∇gΨ, ∗Ψ) − ag(∗Ψ, ∗Ψ)], g and thus we have g(∇ Ψ, ∗Ψ) = ag(∗Ψ, ∗Ψ). A Spin(7) structure is of class U2 if it satisfies ∑4 ∇¯ g¯ − − i+1 ˆ 28( W Φ)(X1,X2,X3,X4) = ( 1) [δΦ(¯p(Xi))Φ(W, X1, .., Xi, .., X4) i=1 (II.9) + 7¯g(W, Xi)δΦ(X1, .., Xˆi, .., X4)]. Suppose W = X1 = ∂r and X2,X3,X4 ∈ TM. For a 3-form ξ on TM we have 3 3 3 3 g¯(¯p(∂r), ξ) =g ¯(∂r, P¯(ξ)) = −Φ(∂r, ξ) = −a r Ψ(ξ) =g ¯(−a r Ψ, ξ) and thusp ¯(∂ ) = −a3r3Ψ. Since ∂ yT¯ = 0 we have ∇¯ g¯ Φ = 0 and the defining relation of the r r ∂r class U2 reduces to 6 6 0 = δΦ(p(∂r))Φ(∂r,X2,X3,X4)+7δΦ(X2,X3,X4) = δΦ(−a r Ψ(X2,X3,X4)Ψ+7X2 ∧X3 ∧X4). 6 6 3 Since a r Ψ(X2,X3,X4)Ψ − 7X2 ∧ X3 ∧ X4 spans Λ (TM) we have δΦ = 0 on TM. For X,Y,Z ∈ TM we have ∑8 ∑7 g¯ 1 g¯ 0 = δΦ(X,Y,Z) = − (∇¯ Φ)(¯ei,X,Y,Z) = − (∇¯ Φ)(ei, X,Y,Z) e¯i a2r2 ei i=1 i=1 ∑7 − 2 2 ∇g − ∇ ∗ = a r [( )ei Ψ](ei,X,Y,Z). i=1 68 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS For X ∈ TM we have ∑8 δΦ(¯p(X)) =δΦ( g¯(¯p(X), e¯i ∧ e¯j ∧ e¯k)¯ei ∧ e¯j ∧ e¯k) i Suppose W = ∂r and X1, .., X4 ∈ TM. Then equation (II.9) gives us ∑4 i+1 3 3 0 = (−1) δΦ(¯p(Xi))a r Ψ(X1, .., Xˆi, .., X4) i=1 ∑4 ∑ 3 3 i = a r (−1) δΨ(el, ej)Ψ(el, ej,Xi)Ψ(X1, .., Xˆi, .., X4). i=1 l For W, Xi ∈ TM, equation (II.9) reduces to ∇¯ g¯ 4 4 ∇g − ∇ ∗ 28( W Φ)(X1,X2,X3,X4) = 28a r [( )W Ψ](X1,X2,X3,X4), which is equal to ∑4 i+1 − (−1) [δΦ(¯p(Xi))Φ(W, X1, .., Xˆi, .., X4) + 7¯g(W, Xi)δΦ(X1, .., Xˆi, .., X4)] i=1 ∑4 ∑ 4 4 i+1 = a r (−1) δΨ(el, ej)Ψ(el, ej,Xi) ∗ Ψ(W, X1, .., Xˆi, .., X4). i=1 l ∑7 ∇g − ∇ ∗ 0 = [( )ei Ψ](ei,X,Y,Z) i=1 can for example be simplified to 0 = g((ΨxY )xZ, δΨxX) + g(ΨxX, (∇gΨxY )xZ) − g(ΨxX, (∗ΨxY )xZ). Another simplification (see Lemma 4.17) will be used in Example 4.18. 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 69 Theorem 4.13. If the Spin(7) structure on the cone M¯ is of class U1, then: • The G2 structure Ψ on M cannot be of class W3 ⊕ W4. • The G2 structure is of class W1 if and only if the Spin(7) structure is integrable. If the structure on M¯ is of class U2, then the structure on M is never of class W1 ⊕ W3. g Proof. Since the relation g(∇ Ψ, ∗Ψ) = 0 defines the class W2 ⊕ W3 ⊕ W4, we conclude the first result directly from Lemma 4.11. Now, assume the G2 structure Ψ is of class W1, i. e. nearly parallel G2 (see [FG82]): 1 ∇gΨ = g(∇gΨ, ∗Ψ) ∗ Ψ. 168 Taking the scalar product with ∗Ψ on both sides leads to 1 g(∇gΨ, ∗Ψ) = g(∇gΨ, ∗Ψ)g(∗Ψ, ∗Ψ). 168 With the Spin(7) structure being of class U1 and the calculation above we get g(∗Ψ, ∗Ψ) = 1 ∗ ∗ ∗ ∗ ∗ ∗ 168 g( Ψ, Ψ)g( Ψ, Ψ) and thus g( Ψ, Ψ) = 168. Therefore, 1 1 ∇gΨ = g(∇gΨ, ∗Ψ) ∗ Ψ = a g(∗Ψ, ∗Ψ) ∗ Ψ = a ∗ Ψ. 168 168 Thus ∇gΨ = ∇Ψ = a ∗ Ψ and with Remark 4.10 we get ∇ = ∇g and ∇¯ g¯ = ∇¯ . Since ∇¯ Φ = 0 the Spin(7) structure on M¯ is integrable. Consider a structure on M¯ of class U2. With Lemma 4.11 we get δΦ = 0 on TM. To see that this structure is integrable it is sufficient to show ∂ryδΦ = 0, see [Fe86]. We have for X,Y ∈ TM ∑8 ∑7 y − ∇¯ g¯ ∇g − ∇ (∂r δΦ)(X,Y ) = ( e¯i Φ)(¯ei, ∂r,X,Y ) = ar (( )ei Ψ)(ei,X,Y ) i=1 i=1 ∑7 =ar [(∇g Ψ)(e ,X,Y ) + Φ(e , e ,X,Y )] = −arδΨ(X,Y ). ei i i i i=1 This is equal to zero if the structure on M is cocalibrated (of class χ1⊕χ3, defined by δΨ = 0). 4.2 Corresponding spinors on G2 manifolds and their cones − c − 2a ¯ − c 2 2 − 2 2 c − 2a 2 2 Since we have T T = 3 Ψ, the difference T T is the lift of a r T a r T = 3 a r Ψ. 1 y ¯ Furthermore, a3r3 ∂r Φ is the lift of Ψ to M, hence we have 2 T¯ − T c = − ∂ yΦ. 3r r Now Lemma 2.4 implies: ∇c 1 Theorem 4.14. For a G2 manifold with characteristic connection and for α = 2 a or α = − 1 2 a, there is 1. a one to one correspondence between Killing spinors with torsion ∇s X ϕ = αXϕ ∇¯ s 4s y ¯ on M, and parallel spinors of the connection + 3r ∂r Φ on M with cone constant a 2s ∇¯ s ϕ + (Xy(∂ yΦ))ϕ = 0. X 3r r 70 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS 2. a one to one correspondence between ∇¯ s-parallel spinors on M¯ with cone constant a and spinors on M satisfying 2as ∇s ϕ = αXϕ + (XyΨ)ϕ. X 3 1 In particular for s = 4 we get the correspondence spinors on M spinors on M¯ ∇c ∇¯ − 1 y y X ϕ = αXϕ X ϕ = 6r (X (∂r Φ))ϕ ∇c a y ∇¯ X ϕ = αXϕ + 6 (X Ψ) ϕ X ϕ = 0 Remark 4.15. As for metric almost contact structures (see Remark 3.17), one can use the ¯ − − 7 ∗ ∧ 1 ∗ ∧ characterisation T = δΦ 6 (θ Φ) with θ = 7 (δΦ Φ) (see [Iv04]) and the description of T c given in Theorem 4.8 of [FI02] to rewrite these equations in terms of the geometric data of the Spin(7) structure. Remark 4.16. In Section 3 of Chapter I we see, that a G2 structure also is given by a spinor ϕ, which is ∇c-parallel. On the other hand, for any Spin(7) manifold there is a spinor that is parallel with respect to the characteristic connection (Theorem 1.1 of [Iv04]). The G2 spinor induces the Spin(7) spinor in the following way: From Lemma 4.2 we know that (XyΨ) · ϕ = −3X · ϕ and thus ∇c · y · X ϕ = 0 = 3X ϕ + (X Ψ) ϕ, which is the identity for a spinor on M inducing a ∇¯ -parallel spinor on M¯ in Theorem 4.14. Be cautious that ∇c may have more parallel spinor fields than just ϕ; for these, we cannot define a suitable ‘lifted’ spinor on the cone, unless one finds a similar trick to write the spinor field equation in a form covered by Theorem 4.14. This remark indicates, that one might use the description of a Spin(7) and a G2 structure via spinors to calculate the correspondences given in Theorem 4.13 as it was done in Section 1 for SU(3) and G2 manifolds. Note that the stabilizer of a spinor in dimension 8 is not always the group Spin(7), so one does not get correspondences as in Lemmas 3.1 and 4.1 of Chapter I. 4.3 Examples To simplify the calculations in the example we reformulate the second condition for a G2 structure on M to imply a Spin(7) structure of class U1 on M¯ of Lemma 4.11. So we only have to calculate Ψ, ∗Ψ and ∇gΨ to check the conditions. Lemma 4.17. The second condition of Lemma 4.11 g(∗Ψ, [(∇ − ∇g) ∗ Ψ]xX) = 3g(Ψ, [(∇ − ∇g)Ψ]xX) is equivalent to [ ∑7 ∗ ∇g 0 = Ψ(ei, ej, ek, el)( ei Ψ)(ej, ek, em)Ψ(em, el,X) i,k,j,l,m=1 g + ∗Ψ(ei, ej, ek, el)(∇ Ψ)(el, X, em)Ψ(em, ej, ek) − ∗Ψ(ei, ej, ek, el) ∗ Ψ(ei, ej, ek, em)Ψ(em, el,X) ei ] − ∗Ψ(ei, el, ej, ek) ∗ Ψ(ei, el, X, em)Ψ(em, ej, ek) [ ] ∑7 + 3 − Ψ(e , e , e )(∇g Ψ)(e , e ,X) + a Ψ(e , e , e ) ∗ Ψ(e , e , e ,X) . i j k ei j k i j k i j k i,k,j=1 4. G2 STRUCTURES – SPIN(7) STRUCTURES ON THE CONE 71 Proof. We continue the calculation of the proof of Lemma 4.11 and with Lemma 4.3 we get ∑7 1 g 0 = (∇ ∗ Ψ)(ej, ek, a ∗ Ψ(ei, el, ej, ek)el,X) a ei i,k,j,l=1 1 ∑7 − (∇ ∗ Ψ)(e , e , a ∗ Ψ(e , e , e , e )e ,X) a ei j k i l j k l i,k,j,l=1 ∑7 ∑7 − 3 Ψ(e , e , e )(∇g Ψ)(e , e ,X) + 3a Ψ(e , e , e ) ∗ Ψ(e , e , e ,X) i j k ei j k i j k i j k i,k,j=1 i,k,j=1 ∑7 = ∗Ψ(e , e , e , e )(∇g ∗ Ψ)(e , e , e ,X) i l j k ei j k l i,k,j,l=1 ∑7 − ∗ ∇ ∗ Ψ(ei, el, ej, ek)( ei Ψ)(ej, ek, el,X) i,k,j,l=1 ∑7 ∑7 − 3 Ψ(e , e , e )(∇g Ψ)(e , e ,X) + 3a Ψ(e , e , e ) ∗ Ψ(e , e , e ,X) i j k ei j k i j k i j k i,k,j=1 i,k,j=1 ∑7 = − ∗Ψ(e , e , e , e )(∇g Ψ)(e , e ,P (e ,X)) i l j k ei j k l i,k,j,l=1 ∑7 − ∗Ψ(e , e , e , e )(∇g Ψ)(e ,X,P (e , e )) i l j k ei l j k i,k,j,l=1 ∑7 ∗ ∇ + Ψ(ei, el, ej, ek)( ei Ψ)(ej, ek,P (el,X)) i,k,j,l=1 ∑7 ∗ ∇ + Ψ(ei, el, ej, ek)( ei Ψ)(el,X,P (ej, ek)) i,k,j,l=1 ∑7 ∑7 − 3 Ψ(e , e , e )(∇g Ψ)(e , e ,X) + 3a Ψ(e , e , e ) ∗ Ψ(e , e , e ,X) i j k ei j k i j k i j k i,k,j=1 i,k,j=1 which is equal to the condition stated in the lemma. Example 4.18. Let (M, ξ1, ξ2, ξ3, η1, η2, η3) be a 7 dimensional 3-Sasaki manifold with corre- sponding 2-forms Fi, i = 1, 2, 3. Let ηi for i = 1, .., 7 be the dual of a local basis {e1 = ξ1, e2 = ξ2, e3 = ξ3, e4, .., e7}, such that F1 = −η23 − η45 − η67,F2 = η13 − η46 + η57,F3 = −η13 − η47 − η56. Here for ηi ∧ .. ∧ ηj we write ηi,..,j. In [AF10] it is explained that there is no characteristic connection as such, but one can construct a cocalibrated G2 structure Ψ = η1 ∧ F1 + η2 ∧ F2 + η3 ∧ F3 + 4η1 ∧ η2 ∧ η3 = η123 − η145 − η167 − η246 + η257 − η347 − η356 c c with characteristic connection ∇ and torsion T = η1 ∧ dη1 + η2 ∧ dη2 + η3 ∧ dη3 that is very well adapted to the 3-Sasakian structure. It is therefore called the canonical G2 structure of the 72 CHAPTER II. HYPERSURFACES, CONES AND GENERALIZED KILLING SPINORS underlying 3-Sasakian structure. Remark 4.16 ensures then the existence of a ∇¯ -parallel spinor field on M¯ . We calculate the class of the Spin(7) structure on M¯ of the canonical G2 structure using Lemma 4.11. Theorem 4.19. The Spin(7) structure on the cone constructed from the canonical G2 structure U 15 of a 3-Sasakian manifold is of class 1 if and only if the cone constant is a = 14 . Proof. Due to the formulation of the second condition of Lemma 4.11 given in Lemma 4.17, we just need to calculate ∗Ψ and ∇gΨ. Obviously ∗Ψ is given by ∗Ψ = η4567 − η2367 − η2345 − η1357 + η1346 − η1256 − η1247. To get ∇gΨ we observe ∇g Ψ = (∇g η ) ∧ F + (∇g η ) ∧ F + (∇g η ) ∧ F ej ej 1 1 ej 2 2 ej 3 3 + η ∧ (∇g F ) + η ∧ (∇g F ) + η ∧ (∇g F ) 1 ej 1 2 ej 2 3 ej 3 + 4(∇g η ) ∧ η ∧ η + 4η ∧ (∇g η ) ∧ η + 4η ∧ η ∧ (∇g η ) ej 1 2 3 1 ej 2 3 1 2 ej 3 and since (η ,F ) are Sasakian structures we have (∇g F )(Y,Z) = g(e ,Z)η (Y )−g(e ,Y )η (Z). i i ej i j i j i Thus ∇g F = η ∧ η for i = 1, 2, 3 and j = 1, .., 7 implying η ∧ (∇g F ) = 0. Since ej i j i i ej i ∇g ∇g − ( X ηi)Y = g(Y, X ξi) = g(Y, ΨiX) = Fi(X,Y ) ∇g y we have X ηi = X Fi and get ∇g Ψ =(e yF ) ∧ F + (e yF ) ∧ F + (e yF ) ∧ F ej j 1 1 j 2 2 j 3 3 + 4(ejyF1) ∧ η2 ∧ η3 + 4η1 ∧ (ejyF2) ∧ η3 + 4η1 ∧ η2 ∧ (ejyF3). This gives us ∇g Ψ = − η + η + η + η , ∇g Ψ = η + η − η − η , e1 346 357 247 256 e2 345 367 147 156 ∇g Ψ = − η − η + η − η , ∇g Ψ = 3 (−η + η + η − η ), e3 245 267 146 157 e4 235 567 136 127 ∇g Ψ = 3 (η − η − η − η ), ∇g Ψ = 3 (−η + η − η + η ), e5 234 467 137 126 e6 237 457 134 125 ∇g Ψ = 3 (η − η + η + η ). e7 236 456 135 124 Using an appropriate computer algebra system we easily calculate g(∇gΨ, ∗Ψ) = 180, g(∗Ψ, ∗Ψ) = 168, 15 thus the first condition of Lemma 4.11 is satisfied if a = 14 . Using the formulation given in Lemma 4.17 of the second condition one easily checks that the this condition is satisfied for any a. 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Geom. 7, 59-68 (1989). 77 Jos H¨oll Weidenh¨auserstr.50 35037 Marburg Lebenslauf Pers¨onliche Daten Name Jos H¨oll Geboren am 13.05.1984 in Herrenberg Ausbildung September 1994 - Juli 2003 Freie Waldorfschule T¨ubingen,Abschluss: Abitur Oktober 2005 - Juli 2007 Studium der Mathematik mit Nebenfach Informatik an der Ruprecht-Karls-Universit¨atHeidelberg, Abschluss des Vordiploms August 2007 - Februar 2008 Auslandssemester im Studium der Mathematik an der Uni- versit¨atKopenhagen M¨arz2008 - Mai 2011 Studium der Mathematik mit Nebenfach Informatik an der Philipps-Universit¨atMarburg Abschluss: Diplom Juni 2011 - Heute Promotion (Mathematik) an der Philipps-Universit¨atMar- burg Beruflicher Werdegang Juni 2011 - Heute Wissenschaftlicher Mitarbeiter an der Universit¨atMarburg, t¨atigin Lehre und Forschung Januar 2012 - Heute Empf¨angereines Promotionsstipendiums der Marburg Uni- versity Research Academy (MARA)