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Summary

In the mathematical field of differential geometry, a Kenmotsu manifold is an almost-contact manifold endowed with a certain kind of Riemannian metric.

Definitions

Let $(M,\varphi ,\xi ,\eta )$ be an almost-contact manifold. One says that a Riemannian metric $g$ on $M$ is adapted to the almost-contact structure $(\varphi ,\xi ,\eta )$ if:

{\begin{aligned}g_{ij}\xi ^{j}&=\eta _{i}\\g_{pq}\varphi _{i}^{p}\varphi _{j}^{q}&=g_{ij}-\eta _{i}\eta _{j}.\end{aligned}} That is to say that, relative to $g_{p},$ the vector $\xi _{p}$ has length one and is orthogonal to $\ker \left(\eta _{p}\right);$ furthermore the restriction of $g_{p}$ to $\ker \left(\eta _{p}\right)$ is a Hermitian metric relative to the almost-complex structure $\varphi _{p}{\big \vert }_{\ker \left(\eta _{p}\right)}.$ One says that $(M,\varphi ,\xi ,\eta ,g)$ is an almost-contact metric manifold.

An almost-contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ is said to be a Kenmotsu manifold if

$\nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.$ 