This article is cited in
5 papers
MATHEMATICS
Kenmotsu manifolds with a zero curvature distribution
A. V. Bukusheva National Research Saratov State
University named after G.N. Chernyshevsky, Saratov, Russian Federation
Abstract:
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.
Keywords:
Kenmotsu manifold, Einstein manifold, Schouten tensor, intrinsic connection, $\mathrm{N}$-Connection.
UDC:
514.76
MSC: 53Ñ15 Received: 21.09.2019
DOI:
10.17223/19988621/64/1