DOCSLIB.ORG

Harmonic Analysis in Vector Bundles Associated to the Rotation and Spin Groups*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Harmonic Analysis in Vector Bundles Associated to the Rotation and Spin Groups* View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 106, 314328 (1992) Harmonic Analysis in Vector Bundles Associated to the Rotation and Spin Groups* THOMAS P. BRANSON Deparlmenr of Mathematics. Linic;ersity of’ lowu, Iowa City, Iowa 52242; Murematisk Insriiur. Odense b%ersirer. DK-5230 Odensr M. Denmark; und Sunderfurschungshereich I70 : Geometric und Analysis. Bunsenstrusse 3-5, D-3400 Giirtingcn. German) Communicured by the Edirors Received March 21, 1990 Let V be any vector bundle over the sphere S” which is associated to the prin- cipal bundle of oriented orthonormal frames, or to that of spin frames. We give an explicit formula for the spectrum. with multiplicity, of the Bochner Laplacian V+V on V, where V is the Riemannian or Riemannian spin connection. In the case of tensor bundles, we also give the spectrum of the Lichnerowin Laplacian. The elTect of natural first-order differential operators on the corresponding eigenspaces is investigated, and all results are illustrated in a detailed treatment of some examples: the bundles of ditTerential forms, spinors, trace-free symmetric tensors, and algebraic Weyl tensors. 1 1992 Academic Press. Inc 1. INTRODUCTION AND PRINCIPAL RESULTS In a recent paper, Folland [Fo] studies the differential form bundles over the sphere S”, using the representation theory of the special orthogonal groups to give explicit formulas for the eigenvalues of the form Laplacians 6d + dS, and for the multiplicities of these eigenvalues. He also gives a geometric description of the eigenspaces, and determines the effect of the first-order operators d and 6 on them. Questions of this type were also treated in [GM, pp. 282, 283; IT, Sects. 4, 63, and, for the case of odd n, in [R, Sect. 33. In this paper, we generalize these results by computing the spectrum of the Bochner Laplacian V*V on any tensor-spinor bundle over S” (Theorem 1.1 below). We also give the spectrum of the Lichnerowicz Laplacian of any tensor bundle (Theorem 2.1), and derive some information about the effect of natural first-order differential l Research partially supported by a University of Iowa Faculty Scholar Award. The author gratefully acknowledges the services of the University of Iowa Center for Advanced Studies. 314 0022- 1236192 $5.00 Copyright c 1992 by Acadcmlc Press. Inc All ngbtr of rcproduchon m any lorm reserved HARMONIC ANALYSIS IN VECTOR BUNDLES 315 operators (generalizing d, 6, and the Dirac operator) on eigensections of our Laplacians. We begin by reviewing some standard material. Given any oriented, Riemannian spin manifold X of dimension n 2 3, one has the principal SO(n)- and Spin(n)-bundles of oriented orthonormal and spin frames. We use P (resp. M) as a common abbreviation for these frame bundles (resp. structure groups). If (V, n) is a finite-dimensional representation of M, one can form the associated uector bundle P x, V; this will be irreducible as an M-bundle if and only if (V, n) is an irreducible representation. The irreducible, finite-dimensional representations of SO(n) are parameterized by dominanr weigh& vectors i. in Z’, I= [n/2], with A, 2 ... 21,209 n odd, (1.1) I., z ... >,).,-I 2 141, n even. The irreducible, finite-dimensional representations of Spin(n) which do not factor through SO(n) are parameterized by vectors 1. E (i + Z)’ satisfying (1.1). The representation corresponding to the dominant weight i. and its associated M-vector bundle will both be called V(J). When writing domi- nant weights, we sometimes omit terminal strings of zeroes, and write, for example, a string of k ones as I,. Both the tangent and cotangent bundles are M-isomorphic to V( I ), and the differential form bundles are Ak+A”-“z,+, v(l,), k -Cn/2. If n is even, there is a splitting A’= A: @A’- into + 1 eigenbundles of (J-r)‘*, where * is the Hodge star operator; these bundles are If n is odd, the bundle of spinors is ,?I zM I’(($),); if n is even, there is a splitting Z = Z+ 0 Z- into positive and negative spinors ( + 1 eigen- bundles of 5 = (J-1)“‘” + IV* y1 . y”, where 7 is the fundamental tensor- spinor and the indices are relative to a local oriented orthonormal frame); here The trace-free, symmetric, covariant p-tensors are just TFSP gM V(p), and the algebraic Weyl tensors (traceless tensors with the symmetries of the Riemann curvature tensor) are V(2,2) 0 V(2, - 2), n = 4. 316 THOMAS P. BRANSON The reducibility of -W when n = 4 may be seen as a consequence of the fact that %” is a subbundle of the symmetric tensor square of A’, which splits as A2 zM V( 1, 1) $ V( 1, - 1) in dimension four. The Riemann curvature tensor of the metric is a section of A0 @ TFS’ 0 ry‘, and the summands are represented respectively by the scalar curvature, the Einstein (trace-free Ricci) tensor, and the Weyl conformal curvature tensor. The natural first-order differential operators to consider in the setting of irreducible bundles are the gradienrs and Diruc operutors [SW], which arise when the covariant derivative V is compressed to act between irreducible bundles. Restricted to V(A), the covariant derivative is an M-equivariant differential operator v:c3c(v(;.))+c~-(v(1)@v(i)). (1.2) The bundle on the right in (1.2) need not be irreducible; in fact, it never is unless ). = (0). Indeed, Ul)O V(j.)zM 0 V(P), (1.3 p . i where p +-+ i if and only if p is a dominant weight and either ( 1.4) or ( 1.5 holds: p=j.+e,, some UE { 1, . I), (1.4) n is odd, i,#O, p =i.. (1.5) Here e,=(O,. ,, 1, O,-,). (See, e.g., [Fe, Theorem 3.41.) Note that the con- dition 1. HP is symmetric in i. and p. Since all summands in (1.3) occur with multiplicity one, it is legitimate to speak of the projection Proj,,: V( 1 )@ V(i) -+ V(n) and the operators D+ = Proj,, VI v(i). Note that the leading symbol of D,, is a,(D,,)(<)=fl Proj,(r@.), where 5 is an indeterminate cotangent vector. Some examples are (up to constant factors) the exterior derivative d and coderivative 6 on differential forms (restricted or projected to eigenspaces of the Hodge * if the middle order is involved), the Dirac operator !$ on C’=(C) if n is odd, the Dirac operators v, : C”(Z, ) + C “(Z T ) if n is even, and the Ahlfors operator S: C “( V( 1)) + C “( V(2)) of elasticity and quasiconformal theory. By con- struction, all gradients are M-equivariant, and invariant under isometries which preserve orientation and spin structure. We now specialize to the case where X is the sphere S” with its standard metric, orientation, and spin structure. To bring the representation theory HARMONIC ANALYSIS IN VECTOR BUNDLES 317 of the rotation and spin groups into the picture, we view A’ as the homogeneous space SO(n + 1 )/SO(n) = Spin(n + 1 )/Spin(n); that is, as K/M, where K is a common abbreviation for SO(n + 1) and Spin(n + 1). The left action of K on A’ is by orientation and spin structure preserving isometries, and the isotropy subgroup at any point is isomorphic to M. Given a dominant M-weight i, let I(i) denote the space of K-finite sections of V(;). Let ?X(K) be the space of dominant K-weights; these, of course, are just gotten by changing n to n + 1 in ( 1.1). Given a E T(K), let W(a) be the corresponding (isomorphism class of) K-module. By Frohenius reciprocity [W, Corollary 5.3.3.61, f(i.) splits into irreducible representa- tions of K according to I‘(i) zK @ m,, W(a), m,,, = dim Hom,+,( W(a), V(j.)). 16 T(K) Otherwise stated, the multiplicity of W(a) as a summand in the K-module T(i.) is the multiplicity of V(%) as a summand in the restriction of W(z) to M. But by the classical branching rule (see, e.g., [Bo, p. 143]), m,,, is either Oor 1,andis 1 ifandonlyifa,-i.,EBand We use a 1 i. as an abbreviation for (1.6); thus f(A) zK @ W(a). (1.7) 112 Since V*V is a K-invariant operator, it is diagonalized by the splitting (1.7). Our main result is: THEOREM 1.1. In ( 1.7), the Bochner Laplacian V *V acts on the W(a) summandas multiplication by the constant (20+a,a).~-(2p+A,i,).,, (1.8) where L= [(n+ 1)/2], I= [n/2], tl,,=(n+ I -2h)/2 for h= 1, . L, and p,, = (n - 2a)/2 for a = I, . 1. The dimension of W(a) is dim W(a)= (1.9) 318 THOMAS P. BRANSON where 6 = x + 9 and n + 1 odd, n + 1 et‘en. If a 1 J. ++ p, but it is not the case that a 1 p, then the jkst-order differential operator D;,, annihilates the W(a) summand in ( 1.7). ProoJ The numbers (20 + r, toWL and C, = (2~ + i., i) R~are, respec- tively, the values of the Casimir elements Cas, and Gas”, of the universal enveloping algebras of f=so(n + 1) and m = so(n) in the modules W(r) and V(J). Thus we would like to prove that (1.10) It will convenient to think of X both as the homogeneous space K/M and as the unit sphere in R” + ‘, with homogeneous coordinates x0, .
Load more

Recommended publications
  • Gsm025-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/025 Selected Titles in This Series 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A.
    [Show full text]
  • Â-Genus and Collapsing
    The Journal of Geometric Analysis Volume 10, Number 3, 2000 A A-Genus and Collapsing By John Lott ABSTRACT. If M is a compact spin manifold, we give relationships between the vanishing of A( M) and the possibility that M can collapse with curvature bounded below. 1. Introduction The purpose of this paper is to extend the following simple lemma. Lemma 1.I. If M is a connectedA closed Riemannian spin manifold of nonnegative sectional curvature with dim(M) > 0, then A(M) = O. Proof Let K denote the sectional curvature of M and let R denote its scalar curvature. Suppose that A(M) (: O. Let D denote the Dirac operator on M. From the Atiyah-Singer index theorem, there is a nonzero spinor field 7t on M such that D~p = 0. From Lichnerowicz's theorem, 0 = IDOl 2 dvol = IV~Pl 2 dvol + ~- I•12 dvol. (1.1) From our assumptions, R > 0. Hence V~p = 0. This implies that I~k F2 is a nonzero constant function on M and so we must also have R = 0. Then as K > 0, we must have K = 0. This implies, from the integral formula for A'(M) [14, p. 231], that A"(M) = O. [] The spin condition is necessary in Lemma 1.l, as can be seen in the case of M = CP 2k. The Ricci-analog of Lemma 1.1 is false, as can be seen in the case of M = K3. Definition 1.2. A connected closed manifold M is almost-nonnegatively-curved if for every E > 0, there is a Riemannian metric g on M such that K(M, g) diam(M, g)2 > -E.
    [Show full text]
  • NON-EXISTENCE of METRIC on Tn with POSITIVE SCALAR
    NON-EXISTENCE OF METRIC ON T n WITH POSITIVE SCALAR CURVATURE CHAO LI Abstract. In this note we present Gromov-Lawson's result on the non-existence of metric on T n with positive scalar curvature. Historical results Scalar curvature is one of the simplest invariants of a Riemannian manifold. In general dimensions, this function (the average of all sectional curvatures at a point) is a weak measure of the local geometry, hence it's suspicious that it has no relation to the global topology of the manifold. In fact, a result of Kazdan-Warner in 1975 states that on a compact manifold of dimension ≥ 3, every smooth function which is negative somewhere, is the scalar curvature of some Riemannian metric. However people know that there are manifolds which carry no metric whose scalar scalar curvature is everywhere positive. The first examples of such manifolds were given in 1962 by Lichnerowicz. It is known that if X is a compact spin manifold and A^ 6= 0 then by Lichnerowicz formula (which will be discussed later) then X doesn't carry any metric with everywhere positive scalar curvature. Note that spin assumption is essential here, since the complex projective plane has positive sectional curvature and non-zero A^-genus. Despite these impressive results, one simple question remained open: Can the torus T n, n ≥ 3, carry a metric of positive scalar curvature? This question was settled for n ≤ 7 by R. Schoen and S. T. Yau. Their method involves the existence of smooth solution of Plateau problem, so the dimension restriction n ≤ 7 is essential.
    [Show full text]
  • Gsm038-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/038 Selected Titles in This Series 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V.
    [Show full text]
  • THE DIRAC OPERATOR 1. First Properties 1.1. Definition. Let X Be A
    THE DIRAC OPERATOR AKHIL MATHEW 1. First properties 1.1. Definition. Let X be a Riemannian manifold. Then the tangent bundle TX is a bundle of real inner product spaces, and we can form the corresponding Clifford bundle Cliff(TX). By definition, the bundle Cliff(TX) is a vector bundle of Z=2-graded R-algebras such that the fiber Cliff(TX)x is just the Clifford algebra Cliff(TxX) (for the inner product structure on TxX). Definition 1. A Clifford module on X is a vector bundle V over X together with an action Cliff(TX) ⊗ V ! V which makes each fiber Vx into a module over the Clifford algebra Cliff(TxX). Suppose now that V is given a connection r. Definition 2. The Dirac operator D on V is defined in local coordinates by sending a section s of V to X Ds = ei:rei s; for feig a local orthonormal frame for TX and the multiplication being Clifford multiplication. It is easy to check that this definition is independent of the coordinate system. A more invariant way of defining it is to consider the composite of differential operators (1) V !r T ∗X ⊗ V ' TX ⊗ V,! Cliff(TX) ⊗ V ! V; where the first map is the connection, the second map is the isomorphism T ∗X ' TX determined by the Riemannian structure, the third map is the natural inclusion, and the fourth map is Clifford multiplication. The Dirac operator is clearly a first-order differential operator. We can work out its symbol fairly easily. The symbol1 of the connection r is ∗ Sym(r)(v; t) = iv ⊗ t; v 2 Vx; t 2 Tx X: All the other differential operators in (1) are just morphisms of vector bundles.
    [Show full text]
  • Introduction
    Introduction Spin Geometry is the hidden facet of Riemannian Geometry. It arises from the representation theory of the special orthogonal group SOn, more precisely, from the spinor representation, a certain representation of the Lie algebra son which is not a representation of SOn. Spinors can always be constructed locally on a given Riemannian manifold, but globally there are topological obstructions for their exis- tence. Spin Geometry lies therefore at the cross-road of several subelds of modern Mathematics. Algebra, Geometry, Topology, and Analysis are subtly interwoven in the theory of spinors, both in their denition and in their applications. Spinors have also greatly inuenced Theoretical Physics, which is nowadays one of the main driv- ing forces fostering their formidable development. The Noncommutative Geometry of Alain Connes has at its core the Dirac operator on spinors. The same Dirac operator is at the heart of the Atiyah–Singer index formula for elliptic operators on compact manifolds, linking in a spectacular way the topology of a manifold to the space of solutions to elliptic equations. Signicantly, the classical Riemann–Roch formula and its generalization by Hirzebruch; the Gauß–Bonnet formula and its extension by Chern; and nally Hirzebruch’s topological signature theorem, provide most of the examples in the index formula, but it was the Dirac operator acting on spinors which turned out to be the keystone of the index formula, both in its formulation and in its subsequent developments. The Dirac operator appears to be the primordial example of an elliptic operator, while spinors, although younger than differential forms or tensors, illustrate once again that aux âmes bien nées, la valeur n’attend point le nombre des années.
    [Show full text]
  • A Generalized Lichnerowicz Formula, the Wodzicki Residue Anti Gravity I
    CPT-94jP.3106 Mannheimer Manuskripte 181 November 1994 A generalized Lichnerowicz formula, the Wodzicki Residue anti Gravity I \ Thomas Ackermann 1 and Jürgen \Tolksdorf2 * 1 Fakultät für Mathematik & Informatik, uJrersität Mannheim D-68159 Mannheim, F.R.G 2 Centre de Physique Theorique, CNRS :u.,uminy, Case 907 F-13288 Marseille Cedex 9, France Abstract. We prove a generalized version of the well-known Lichnerowicz formula for the square of_the most general Dirac operator D on hn even-dimensional spin manifold associated to a met~c connection V. We use \thiS form~a to compute 2 the subleading term <I>1(x,x,D2) of the heat-kernel expfl'nsion of D . The trace of this term plays a key-röle in the definitio,n of a (euclidian) gravity action in the context of non-commutative geometry. We show that tlLs gravity action can be interpreted as defining a modified euclidian Einstein-Cattan theory. Keywords: non-commuiaiive geomeiry, Lichnerowicz formula, Dirae operaior, heai-kernel- expansion, Wodzicki residue, graviiy 1 e-mail: [email protected] 2 e-mail: [email protected] ERB 401GT 930224 * Supported by the European Community, contract .~~. Zugangsnummer: ~ S~ \ ~ ~ Signatur: UNIVEnS!T/"T ~./i/J..NNHEIM BibI. cL rsi~.f. Mathematik u. Informatik 1. Introduction . \ When attempting to quantize the electron in 1928, Dirac introduced a first-order operator the square of which is the so-called wav~-operator (d' Alembertian oper- ator). Lateron, in the hands of mathematicians keneralizations of this operator, called 'Dirac operators' - evolved into an importaht tool of modern mathematics, occuring for example in index theory, gauge theodl' geometric quantisation etc.
    [Show full text]
  • Principal Bundles, Vector Bundles and Connections
    Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/.
    [Show full text]
  • Horizontal Dirac Operators in CR Geometry
    Horizontal Dirac Operators in CR Geometry DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) im Fach Mathematik eingereicht an der Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universitat¨ zu Berlin von Dipl.-Math. Christoph Martin Stadtm¨uller Pr¨asidentin der Humboldt-Universit¨atzu Berlin Prof. Dr.-Ing. Dr. Sabine Kunst Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke Gutachter 1. Prof. Dr. Helga Baum 2. Prof. Dr. Uwe Semmelmann 3. Prof. Dr. Bernd Ammann Tag der m¨undlichen Pr¨ufung:14. Juli 2017 Contents Introduction v 1 Contact and CR manifolds and their adapted connections 1 1.1 Contact manifolds . 1 1.2 CR manifolds . 13 1.3 Differential forms on metric contact manifolds . 19 1.3.1 The forms over the contact distribution . 21 1.3.2 The other parts . 23 1.3.3 Application: K¨ahlerform and Nijenhuis tensor . 24 1.4 Adapted connections . 27 1.4.1 Definition and basic properties . 28 1.4.2 The torsion tensor of an adapted connection . 29 1.4.3 The Tanaka-Webster connection and CR connections . 34 1.4.4 Adapted connections with skew-symmetric torsion . 36 1.5 The curvature tensors of an adapted connection . 37 2 Spin geometry on metric contact and CR manifolds 43 2.1 Crash course spin geometry . 43 2.1.1 Clifford algebras and the spin group . 43 2.1.2 Spin manifolds . 47 2.2 Connections and Dirac operators . 49 2.3 The spinor bundle over metric contact manifolds . 54 3 Horizontal Dirac operators 59 3.1 Definition and basic properties .
    [Show full text]
  • Eigenvalue Estimate of the Dirac Operator and Rigidity of Poincare-Einstein Metrics
    EIGENVALUE ESTIMATE OF THE DIRAC OPERATOR AND RIGIDITY OF POINCARE-EINSTEIN METRICS DAGUANG CHEN†, FANG WANG‡ AND XIAO ZHANG♭ Abstract. We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any C3,α conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its lowest value, and any C3,α conformal compactification of some Poincare-Einstein metric must be the flat ball in Euclidean space when the first positive eigenvalue of the boundary Dirac operator achieves certain value relating to the second Yamabe invariant. In two cases the Poincare-Einstein metrics are standard hyperbolic metric. 1. Introduction Let (M, g) be a closed (compact without boundary) n-dimensional Riemannian spin manifold with the positive scalar curvature R > 0. Let λ(D) be the eigenvalue of the Dirac operator D. In 1963, Lichnerowicz [25] firstly proved 1 λ2(D) > inf R. (1.1) 4 M By modifying the Riemammian spin connection suitably, Friedrich [13] improved the Lich- nerowicz inequality (1.1) and obtained the sharp estimate n λ2(D) infR. (1.2) ≥ 4(n 1) M − If the equality holds in (1.2), the manifold is Einstein. In 1986, using conformal covariance of the Dirac operator, Hijazi [16] showed, for n 3, arXiv:1808.02816v2 [math.DG] 3 Dec 2018 n≥ λ2(D) µ , (1.3) ≥ 4(n 1) 1 − where µ1 is the first eigenvalue of the conformal Laplace operator.
    [Show full text]
  • Kähler Manifolds with Small Eigenvalues of the Dirac Operator and a Conjecture of Lichnerowicz Annales De L’Institut Fourier, Tome 49, No 5 (1999), P
    ANNALES DE L’INSTITUT FOURIER ANDREI MOROIANU Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz Annales de l’institut Fourier, tome 49, no 5 (1999), p. 1637-1659 <http://www.numdam.org/item?id=AIF_1999__49_5_1637_0> © Annales de l’institut Fourier, 1999, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 49, 5 (1999), 1637-1659 KAHLER MANIFOLDS WITH SMALL EIGENVALUES OF THE DIRAC OPERATOR AND A CONJECTURE OF LICHNEROWICZ by Andrei MOROIANU 1. Introduction. The problem of finding optimal lower bounds for the eigenvalues of the Dirac operator on compact manifolds was for the first time considered in 1980 by Th. Friedrich [3]. Using the Lichnerowicz formula [19] and a modified spin connection, he proved that the first eigenvalue A of the Dirac operator on a compact spin manifold (M71,?) of positive scalar curvature S satisfies (1) A2^————inf5. 4(n — 1) M The limiting case of this inequality is characterized by the existence of real Killing spinors. After several partial results of Th. Friedrich, I. Kath, R. Grunewald, and 0. Hijazi ([12], [6], [7], [8], [12]), the geometrical description of simply connected manifolds carrying Killing spinors was obtained in 1991 by C.
    [Show full text]
  • Erwin Schrödinger, Dirac Electron in the Gravitational Field I
    This is a repository copy of Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field I. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/154944/ Version: Accepted Version Article: Kay, Bernard S. orcid.org/0000-0002-8766-0532 (2020) Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field I. General Relativity and Gravitation. ISSN 0001-7701 https://doi.org/10.1007/s10714-019-2625-z Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Noname manuscript No. (will be inserted by the editor) Editorial note to: Erwin Schr¨odinger, Dirac electron in the gravitational field I Bernard S. Kay Received: date / Accepted: date Abstract We aim to give a mathematical and historical introduction to the 1932 paper “Dirac equation in the gravitational field I” by Erwin Schr¨odinger on the generalization of the Dirac equation to a curved spacetime and also to discuss the influence this paper had on subsequent work.
    [Show full text]