On geometry of Kenmotsu manifolds with N-connection

Abstract

A Kenmotsu manifold with a given N-connection is considered. From the integrability of the distribution of a Kenmotsu manifold it follows that the N-connection belongs to the class of the quarter-symmetric connections. Among the N-connections, the class of connections adapted to the structure of the Kenmotsu manifold is specified. In particular, it is proved that an N-connection preserves the structure endomorphism φ of the Kenmotsu manifold if and only if the endomorphisms N and φ commute. A formula expressing the N-connection in terms of the Levi-Civita connection is obtained. The Chrystoffel symbols of the Levi-Civita connection and of the N-connection of the Kenmotsu manifold with respect to the adapted coordinates are computed. The properties of the invariants of the interior geometry of the Kenmotsu manifolds are investigated. The invariants of the interior geometry are the following: the Schouten curvature tensor; the 1-form  defining the distribution D; the Lie derivative 0   L g of the metric tensor g along the vector field ;  the tensor field P with the components given with respect to the adapted coordinate system by the formula Pacd  ncad . The field P is called in the work the Schouten — Wagner tensor. It is proved that the Schouten — Wagner tensor of the interior connection of the Kenmotsu manifold is zero. The conditions that satisfies the endomorphism N defining the metric N-connection are found. At the end of the work, an example of a Kenmotsu manifold with a metric N-connection preserving the structure endomorphism φ is given.