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 . 59 3.2 Weitzenb¨ock and Schr¨odinger-Lichnerowicz type formulae . 69 3.3 CR-conformal covariance . 82 3.4 Example: S1-bundles . 87 η 3.4.1 Application: the spectrum of DH on spheres . 92 3.5 Example: Compact quotients of the Heisenberg group . 101 4 Analysis of the horizontal Dirac Operators: Heisenberg Calculus 115 4.1 Introduction . 115 4.2 Heisenberg manifolds and their tangent groups . 117 4.2.1 Coordinates adapted to Heisenberg manifolds . 120 iii Introduction 4.3 Heisenberg order and Heisenberg symbol of differential operators . 124 4.4 Heisenberg pseudodifferential operators . 129 4.4.1 Invariant pseudodifferential operators on the tangent group . 131 4.4.2 Pseudodifferential operators { local theory . 139 4.4.3 Heisenberg pseudodifferential operators on manifolds . 145 4.5 Hypoellipticity of H-differential operators . 151 4.5.1 Rockland operators . 151 4.5.2 Hypoellipticity of Heisenberg differential operators . 153 4.5.3 Spectrum of Heisenberg-elliptic differential operators . 155 4.5.4 Operators of Sublaplace type on CR manifolds . 160 4.5.5 The Tanaka-Webster operator on the extremal bundles . 165 Appendix: Some functional analysis 171 A.1 Test functions, distributions and the Fourier transform . 171 A.2 Fredholm operators . 175 A.3 Sobolev spaces . 176 iv Introduction 2m+1 m+1 Consider a hypersurface M of real codimension one in C or, more generally, in a complex manifold. In particular, such a hypersurface may arise as the boundary m+1 of a complex domain Ω ⊂ C . Then, its tangent space TpM at any p 2 M cannot be stable under multiplication with the imaginary unit i as it is of odd real dimension. Its largest subspace that is stable under i is given as Hp = TpM \i·TpM. Then, the multiplication with i descends to an isomorphism Jp : Hp ! Hp such that 2 Jp = − IdHp and the integrability conditions [JX; Y ] + [X; JY ] 2 Γ(H) [JX; JY ] − [X; Y ] − J([JX; Y ] + [X; JY ]) = 0 for all X; Y 2 Γ(H) are satisfied. This is the standard model for CR manifolds and an abstract CR manifold is then defined as an odd-dimensional manifold M 2m+1 together with a distribution H ⊂ TM of rank 2m and a fibrewise isomorphism J : H ! H that satisfy the conditions above. Let it be noted that more general notions of CR manifolds allowing higher codimension of H and not requiring inte- grability of J exist, but we will content ourselves with the more restrictive definition above. The oldest results on what has since developed into CR geometry are due to 2 Poincar´e[Poi07] who showed that two real hypersurfaces of C are generally not biholomorphically equivalent. This was generalised to higher dimensions by Chern and Moser [CM74] and N. Tanaka [Tan62]. Much of the early literature on CR geometry is decidedly analytic in nature, discussing certain PDEs on CR manifolds, for instance [Koh63, KR65, Koh65]. An more geometrically flavoured investigation began with the works of S.M. Web- ster [Web78] and N. Tanaka [Tan75] introducing a metric and a connection on CR- manifolds as follows: If M is oriented, then there exists a (non-unique) one-form η 2 Ω1(M) such that H = ker η. To such a form, one associates the Levi form 2 ∗ Lη 2 Γ(Sym H ) via 1 Lη(X; Y ) = 2 dη(X; JY ): If Lη is positive-definite, we obtain a metric gη on M, the so-called Webster metric defined by gη = Lη + η ⊗ η; v Introduction which allows us to do geometry in the Riemannian sense on M. In this case, (M; H; J; η) is called a strictly pseudoconvex CR manifold. On such a manifold, m η is in fact a contact form, i.e. it satisfies η ^ (dη) 6= 0, and moreover, gη is com- patible with η and J in the sense that 2gη(JX; Y ) = dη(X; Y ), where we extend J by zero on H?. This includes strictly pseudoconvex CR manifolds in the larger class of metric contact manifolds, i.e. tuples (M; g; η; J) of a Riemannian metric g on M, 1 η 2 Ω (M) and J 2 End(M) such that kηkg = 1 (pointwise), 2g(JX; Y ) = dη(X; Y ) and J 2 = − Id +η ⊗ η]. Note that the Riemannian structure on a CR manifold is unique only up to the choice of η which can be conformally changed by any function u 2 C1(M) to obtain a new strictly pseudoconvex structure formη ~ = e2uη. The main ingredient in Riemannian geometry beside a metric g is a connection r. In particular, if one has additional geometric objects on the manifold, one would like them to be parallel under the connection. In the case of a strictly pseudoconvex CR manifold, J will never be parallel under the Levi-Civita connection rg (unlike in the almost-Hermitian case, where J is parallel under rg in the subclass of K¨ahlerman- ifolds). Thus, we replace rg with an adapted connection, i.e. a metric connection that parallelises J (and thus η). The most well-known of these adapted connections is the Tanaka-Webster rη connection developed independently by N. Tanaka and S.M. Webster that is defined through its torsion, which is given by 1 T (X; Y ) = dη(X; Y )ξ and T (ξ; X) = 2 ([X; ξ] + J[JX; ξ]) for X; Y 2 Γ(H), where ξ is the Reeb vector field characterised equivalently as the metric dual of η or by η(ξ) = 1 and ξydη = 0. While the Tanaka-Webster connection is used in most work on CR manifolds, the space of adapted connections is much larger. Some examples of adapted connections have been constructed by L. Nicolaescu [Nic05] (through Hermitian connections on M × R) and (in the five-dimensional case) by C. Puhle [Puh11]. It is then a nat- ural question how large the space of adapted connections is and how they can be characterised. In the almost-Hermitian case (which, to some extent, is the even- dimensional analogue of metric contact manifolds) such a classification of Hermitian connections is available, it is due to P. Libermann [Lib54] (compare also the dis- cussion in modern language by P. Gauduchon [Gau97]). In this thesis, we give an explicit description of the class of adapted connections on a general metric contact manifold. Any metric connection is defined by its torsion tensor and through a care- ful decomposition of the space Ω2(M; T M), we can explicitly describe the space of torsion tensors of adapted connections (cf Theorem 1.4.3). Theorem. Let (M; g; η; J) be a metric contact manifold and r an adapted connec- tion. Then its torsion tensor has the following form: 0;2 9 3 1 T = N + 8 ! − 8 M! + B + ξ ⊗ dη − 2 η ^ (JJ ) + η ^ Φ; vi Introduction where ! is a three-form on H whose decomposition into (p; q)-forms consists only of forms of type (2; 1) and (1; 2), B 2 Ω2(H; H) satisfies B(J·;J·) = B and vanishes under the Bianchi operator and Φ is a skew-symmetric endomorphism of H satisfying ΦJ = JΦ. The other parts are completely determined by the geometry of the contact structure. Conversely, given any !; B; Φ as above there exists exactly one adapted connection r(!; B; Φ) whose torsion is as given above. In this description, the Tanaka-Webster connection is the connection obtained by setting all freely choosable parts to zero. We would now like to do spin geometry over CR manifolds. In Riemannian man- ifolds, a lot of geometric information is reflected in the spin Dirac operator of the manifold and the hope is that the CR geometry is similarly reflected in an appropri- ately chosen Dirac operator. Beside this interest for the possible information about the CR geometry itself, this is also of interest in Lorentzian geometry. Through the Feffermann construction (originally described by C. Feffermann [Fef76] for products 1 m+1 @Ω × S where Ω ⊂ C is a complex domain), one obtains a Lorentzian metric Fη on S1-bundles over a strictly pseudoconvex CR manifold (M; H; J; η).