On infinitesimal conformal and projective transformations of normal contact spaces

Abstract

Introduction. In the previous paper [ 4 ]° the author discussed some properties of normal contact spaces. However, problems concerning infinitesimal transformations have not been studied. In the present paper such problems are concerned and some satisfactory answers are given. In § 1, we state the fundamental identities of normal contact spaces. In § 2, we shall give some preliminary facts concerning infinitesimal transformations for the later use. After these preparations, in § 3, v e shall prove that an infinitesimal conformal transformation in normal contact spaces is necessarily concircular. Thus we know that a connected complete normal contact space admitting an infinitesimal non-isometric conformal transformation is isometric with a unit sphere. It will be shown in § 4 that an infinitesimal projective transformation in a normal contact space has some analogous properties of the one in an Einstein space, for example, that any infinitesimal projective transformation in this space is decomposed as a sum of a Killing vector and an infinitesimal gradient projective transformation. In § 5, we shall define the notion of ^-Einstein spaces and discuss infinitesimal conformal and projective transformations in such spaces. Finally, we shall devote § 6 to show that one of Sasaki's examples of normal contact spaces is an example of ^-Einstein spaces. l (Φ>?> v> ^-structure and contact structure. On an n(= 2m + l)-dimensional real differentiable manifold M with local coordinate systems {x}, if there exist a tensor field φ/, contravariant and covariant vector fields ξ and ηt satisfying the relations (1. 1) ξ% = 1, (1. 2) rank | φ / | = n 1, (1. 3) φ/ξ> = 0, (1. 4) φ/Vί = 0, (15) Φ / Φ * ' = δ * < + *7*f,