Kenmotsu Manifolds with Zero Ricci-Schouten Tensor

Abstract

The paper is dedicated to the investigation of the interior geometry of the Kenmotsu manifolds M. By the interior geometry of the manifold M we mean the aggregate of the properties of the manifold that depend only on the closing of the distribution D of the Kenmotsu manifold as well as on the parallel transport of the vectors from the distribution D along arbitrary curves of the manifold. The invariants of the interior geometry of a Kenmotsu manifold are the following: the Schouten curvature tensor; the 1-form η generating the distribution D; the Lie derivative of the metric tensor g along the structure vector field ; the Schouten-Wagner admissible tensor fields with the components with respect to adapted coordinates; the structural endomorphism φ; the endomorphism N that allows to prolong the interior connection to a connection in a vector bundle. A special attention is payed to the Ricci-Schouten tensor. In particular, it is stated that a Kenmotsu manifold with zero Ricci-Schouten tensor is an Einstein manifold. Conversely, if M is an η-Einstein Kenmotsu manifold and then M is an Einstein manifold with zero Ricci-Schouten tensor. It is proved that the Ricci-Schouten tensor is zero if and only if the Kenmotsu manifold M is locally Ricci-symmetric. This implies the following well-known result: a Kenmotsu manifold is an Einstein manifold if and only if it is locally Ricci-symmetric. An N-connection with torsion, is introduced; this connection is Ricci-flat if and only if M is an Einstein manifold.