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Homogeneous Almost Hypercomplex and Almost Quaternionic Pseudo-Hermitian Manifolds with Irreducible Isotropy Groups

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Homogeneous Almost Hypercomplex and Almost Quaternionic Pseudo-Hermitian Manifolds with Irreducible Isotropy Groups Homogeneous almost hypercomplex and almost quaternionic pseudo-Hermitian manifolds with irreducible isotropy groups Dissertation zur Erlangung des Doktorgrades der Fakultät für Mathematik, Informatik und Naturwissenschaften der Universität Hamburg vorgelegt im Fachbereich Mathematik von Benedict Meinke Hamburg 2015 Datum der Disputation: 18.12.2015 Folgende Gutachter empfehlen die Annahme der Dissertation: Prof. Dr. Vicente Cortés Suárez Prof. Dr. Ines Kath Prof. Dr. Abdelghani Zeghib Contents Acknowledgements v Notation and conventions ix 1 Introduction 1 2 Basic Concepts 5 2.1 Homogeneous and symmetric spaces . .5 2.2 Hyper-Kähler and quaternionic Kähler manifolds . .9 2.3 Amenable groups . 11 2.4 Hyperbolic spaces . 15 2.5 The Zariski topology . 21 3 Algebraic results 23 3.1 SO0(1; n)-invariant forms . 23 3.2 Connected H-irreducible Lie subgroups of Sp(1; n) .............. 34 4 Main results 41 4.1 Classification of isotropy H-irreducible almost hypercomplex pseudo-Hermitian manifolds of index 4 . 41 4.2 Homogeneous almost quaternionic pseudo-Hermitian manifolds of index 4 with H-irreducible isotropy group . 45 5 Open problems 47 A Facts about Lie groups and Lie algebras 49 B Spaces with constant quaternionic sectional curvature 51 Bibliography 55 iii Acknowledgements First of all I would like to thank my advisor Vicente Cort´es for suggesting the topic and his patient mentoring, continuous encouragement and support, and helpful discussions during the last four years. Special thanks go to Abdelghani Zeghib and the ENS de Lyon for helpful discussions and hospitality during my stay in Lyon in October 2013. I would like to thank Marco Freibert for carefully reading my thesis and giving me valu- able feedback. I also thank his wife Antonia Freibert for baking me birthday cookies and sending them to me from Denmark. Thanks also go to Christoph Wockel who helped me find references and to David Lindemann and Reiner Lauterbach for helpful discussions. Furthermore, I would like to express my gratitude to my office mate Owen Vaughan and to our secretary Eva Kuhlmann for the friendly atmosphere in the department. Finally, I want to thank my friends and my family for their support. v Notation and conventions We give here a collection of the notation used in this thesis. • Numbers and vector spaces: We denote by R the set of real numbers, by C the set of complex numbers, and by H the set of quaternions. We will refer to all of these sets as fields, although H is a skew-field. Let R+ be the set of real positive numbers. We denote by i the imaginary unit of C. Let i, j, and k be quaternions that satisfy the quaternionic relations, 2 2 2 namely i = j = k = −1 and ij = −ji = k. We will write F if we do not want to specify any of the three fields R, C, or H. For x 2 F we denote by <(x) and =(x) the real and the imaginary part of x, respectively. For the purely imaginary I subspace =(H) = spanR fi; j; kg of H we will sometimes use the symbol . If we 0 ∗ speak of a subfield F ⊂ F we mean one of the fields above. Let F := F n f0g be the multiplicative group of F. For an element x 2 F we denote by x the conjugate of x p and by kxk := xx the norm of x. Let N be the set of natural numbers. The letter n will always denote a natural n number. We will denote by F the (right) vector space of n-tuples of elements of F. n n Let e1; : : : ; en be the standard basis of F . An F-subspace of F is a subspace over 0 0 n F. Analogously, if F ⊂ F is a subfield, then an F -subfield of F is a subspace of n 0 n Pn F over F . Let h·; ·i0 be the standard inner product on F , i.e. hx; yi0 = i=1 xiyi. n n n Let B (F) = fv 2 F j hv; vi0 < 1g be the open ball of radius 1 and let S (F) = n fv 2 F j hv; vi0 = 1g be the sphere. For F = C we denote by !0 the Kähler form of 0 0 0 0 0 0 n h·; ·i0 defined by !0(x + iy; x + iy ) := hx; y i0 − hy; x i0 where x; x ; y; y 2 R . k;n n+k Let k be a natural number. Then F denotes the space F endowed with the Pk Pn+k non-degenerate bilinear form h·; ·ik defined by hx; yik := − i=1 xiyi + i=k+1 xiyi. n k;n Let Sk = v 2 F j hv; vik = 1 be the pseudo-sphere. For F = C we define as above 0 0 0 0 the Kähler form !k of h·; ·ik by !k(x + iy; x + iy ) := hx; y ik − hy; x ik. We call a k;n vector v 2 F spacelike, if hv; vik > 0, lightlike, if hv; vik = 0, and timelike, if 1;n hv; vik < 0. For F we will denote by e0; e1; : : : ; en the standard basis where e0 is a timelike vector. k;n We call an F-subspace V of F spacelike, if h·; ·ik restricted to V × V is positive definite, and timelike if the restriction is non-degenerate and V contains timelike vectors. If V is space- or timelike, then we denote by V? its orthogonal complement vii k;n in F with respect to h·; ·ik. k ∗ Let F = R or C and V be a vector space over F. Then Λ V denotes the vector space k ∗ of k-forms on V with values in F. If α 2 Λ V and U ⊂ V is an F-subspace, then k1 ∗ k2 ∗ we denote the restriction of α to U × ::: × U by αjU . For α 2 Λ V , β 2 Λ V we denote by α ^ β 2 Λk1+k2 V∗ the wedge-product. Let α 2 ΛkV∗ and x 2 V, then we denote by ιxα the interior product. • Lie groups and Lie algebras: We denote by G, H, K, and U Lie groups and the associated Lie algebras by g, h, k, and u, respectively. The connected components of the identity are denoted by G0, H0, K0, and U 0. Z(G) is the center of G. For a Lie group G let µ be the Haar measure of G. Let f : G ! C be a Borel- measurable function. Then we define the essential supremum and essential infimum by ess supf := inf fa 2 R j µ(fg 2 Gj jf(g)j > ag) = 0g ; ess inff := sup fb 2 R j µ(fg 2 Gj jf(g)j < bg) = 0g : Notice that kfk1 := ess supf is a semi-norm on the vector space 1 L (G) := ff : G ! C j f Borel-measurable; kfk1 < 1g : We define N := ff : G ! C j f Borel-measurable; kfk1 = 0g. Then we obtain a Banach space by defining L1(G) := L1(G)=N . Let M(n; F) be the set of n × n-matrices with elements of F in the entries and let 1n be the n × n-identity matrix. For a matrix A 2 M(n; F) we denote by A the conjugate matrix and by AT its transpose. We denote by GL(n; F) := fA 2 M(n; F) j A is invertibleg the general linear group. If F = C or R, we denote by SL(n; F) := fA 2 GL(n; F) j det A = 1g the special linear group. Let n U(n; F) := fA 2 GL(n; F) j hAv; Awi0 = hv; wi0 8v; w 2 F g be the Lie group of unitary transformations with respect to h·; ·i0. Equivalently, T U(n; F) consists of the the matrices A which satisfy AA = 1n. If we specify the field F, we will use also the notation Sp(n) := U(n; H); U(n) := U(n; C); O(n) := U(n; R); SU(n) := U(n) \ SL(n; C); SO(n) := O(n) \ SL(n; R): n Sometimes Sp(1) denotes also the Lie group which acts on H by multiplication of quaternions of norm one from the right. Analogously, we define for p; q 2 N the Lie group n p;qo U(p; q; F) := A 2 GL(p + q; F) j hAv; Awip = hv; wip 8v; w 2 F : T Equivalently, U(p; q; F) consists of the matrices A which satisfy A1p;qA = 1p;q where ! −1p 0 1p;q := : 0 1q The maximal Lie subgroup of U(p; q; F) is U(p; F) × U(q; F). If we specify the field F, we will use the following notation Sp(p; q) := U(p; q; H); U(p; q) := U(p; q; C); O(p; q) := U(p; q; R); SU(p; q) := U(p; q) \ SL(p + q; C); SO(p; q) := O(p; q) \ SL(p + q; R): Furthermore, SO0(p; q) denotes the connected component of the identity of SO(p; q). The quaternionic Heisenberg group will be denoted by Heisn(H) and consists of the matrices of the form 0 T 1 2 1 1 −t − 2 ktk + s B C B0 1 t C @ n−1 A 0 0 1 n−1 2 where t 2 H , ktk = ht; ti0, s 2 =(H). • Manifolds: We denote by M always a finite dimensional manifold. If M is connected, then M~ denotes its universal cover. If p 2 M, then TpM is the tangent space of M at p, TM the tangent bundle and End(TM) the endomorphism bundle.
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