Kenmotsu manifolds with a zero curvature distribution

A. V. Bukusheva

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 64 (2020), pp. 5-14

Résumé

We study the Kenmotsu manifold with the intrinsic connection defined on it with the zero Schouten curvature tensor. The Schouten tensor is a nonholonomic analog of the curvature tensor of a Riemannian manifold. The intrinsic connection defines a parallel displacement of admissible vectors along admissible curves. In the case of the zero Schouten tensor, the parallel displacement does not depend on the choice of an admissible curve. It is proved that the Kenmotsu manifold with the zero Schouten tensor is an Einstein manifold. On a Kenmotsu manifold $M$, an $\mathrm{N}$-connectedness is defined, where $\mathrm{N}$ is a tangent bundle endomorphism preserving the distribution $\mathrm{D}$ of the manifold $M$. In some cases, the $\mathrm{N}$-connection is preferable to the Levi–Civita connection. The advantage of $\mathrm{N}$-connectivity is that it satisfies the following condition: $\nabla_{\vec{x}}^N\vec{y}\in\Gamma(D)$, where $\vec{x}\in\Gamma(TM)$, $\vec{y}, \vec{z}\in\Gamma(D)$. In the case when $\mathrm{N=C}$, or $\mathrm{N=C}-\varphi$, $\mathrm{N}$-connectivity $\nabla^N$ coincides with the Tanaka–Webster connection or the Schouten–van Kampen connection, respectively. It is proved that the $\mathrm{N}$-connection curvature tensor is zero if and only if the endomorphism $\mathrm{N}$ is covariantly constant with respect to the intrinsic connection. The covariantly constant with respect to the interior connection of tensor fields can be attributed, in particular, the structural endomorphism $\varphi$ of the Kenmotsu manifold. The interior invariants of the Kenmotsu manifold are investigated. In particular, it is proved that the Schouten–Wagner tensor for the Kenmotsu manifold vanishes. On a distribution $D$ of the Kenmotsu manifold, an almost contact metric structure called the extended structure is determined for the case $\mathrm{N}=\varphi$ by means of $\mathrm{N}$-connection $\nabla^N$. It is proved that in the case of a Kenmotsu manifold with a Schouten tensor, the extended structure is a Kenmotsu structure.

Détail

Keywords: Kenmotsu manifold, Einstein manifold, intrinsic connection, $\mathrm{N}$-Connection.
Mots-clés : Schouten tensor