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 K¨ ahler manifolds are studied in [CL1, CL2, LZ] in

a different context, namely in connection with minimal surfaces and quaternionic

maps between quaternionic K¨ ahler manifolds. The notion of hypercomplex contact

structures will appear in this section again since on such manifolds the real part of