4 1. INTRODUCTION In addition to the above results, in Theorem 7.10 we show that the above con- ditions are equivalent to the property that every Reeb vector field, defined in (2.10), is an infinitesimal generator of a conformal quaternionic contact automorphism, cf. Definition 7.7. Finally, we also develop useful tools necessary for the geometry and analysis on QC manifolds. We define and study some special functions, which will be relevant in the geometric analysis on quaternionic contact and hypercomplex manifolds as well as properties of infinitesimal automorphisms of QC structures. In particular, the considered anti-regular functions will be relevant in the study of qc-pseudo-Einstein structures, cf. Definition 6.1. Organization of the paper: In the following two chapters we describe in details the notion of a quaternionic contact manifold, abbreviate sometimes to QC- manifold, and the Biquard connection, which is central to the paper. In Chapter 4 we write explicitly the Bianchi identities and derive a system of equations satisfied by the divergences of some important tensors. As a result we are able to show that qc-Einstein manifolds, i.e., manifolds for which the restriction to the horizontal space of the qc-Ricci tensor is proportional to the metric, have constant scalar curvature, see Theorem 4.9. The proof uses Theorem 4.8 in which we derive a relation between the horizontal divergences of certain Sp(n)Sp(1)-invariant tensors. By introducing an integrability condition on the horizontal bundle we define hyperhermitian contact structures, see Definition 4.14, and with the help of Theorem 4.8 we prove Theorem 1.3. Chapter 5 describes the conformal transformations preserving the qc-Einstein condition. Note that here a conformal quaternionic contact transformation between two quaternionic contact manifold is a diffeomorphism Φ which satisfies Φ∗η = μ Ψ · η, for some positive smooth function μ and some matrix Ψ SO(3) with smooth functions as entries and η = (η1,η2,η3)t is considered as an element of R3. One defines in an obvious manner a point-wise conformal transformation. Let us note that the Biquard connection does not change under rotations as above, i.e., the Biquard connection of Ψ·η and η coincides. In particular, when studying conformal transformations we can consider only transformations with Φ∗η = μ η. We find all conformal transformations preserving the qc-Einstein condition on the quaternionis Heisenberg group or, equivalently, on the quaternionic sphere with their standard contact quaternionic structures proving Theorem 1.1. Chapter 6 concerns a special class of functions, which we call anti-regular, defined respectively on the quaternionic space, real hyper-surface in it, or on a quaternionic contact manifold, cf. Definitions 6.6 and 6.15 as functions preserving the quaternionic structure. The anti-regular functions play a role somewhat similar to those played by the CR functions, but the analogy is not complete. The real parts of such functions will be also of interest in connection with conformal transformation preserving the qc-Einstein tensor and should be thought of as generalization of pluriharmonic functions. Let us stress explicitly that regular quaternionic functions have been studied extensively, see [S] and many subsequent papers, but they are not as relevant for the considered geometrical structures. Anti-regular functions on hyperk¨ ahler and quaternionic ahler manifolds are studied in [CL1, CL2, LZ] in a different context, namely in connection with minimal surfaces and quaternionic maps between quaternionic ahler manifolds. The notion of hypercomplex contact structures will appear in this section again since on such manifolds the real part of
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