Almost-contact manifold
In the mathematical field of differential geometry, a Kenmotsu manifold is an almost-contact manifold endowed with a certain kind of Riemannian metric. They are named after the Japanese mathematician Katsuei Kenmotsu.
Definitions
Let
be an almost-contact manifold. One says that a Riemannian metric
on
is adapted to the almost-contact structure
if:
![{\displaystyle {\begin{aligned}g_{ij}\xi ^{j}&=\eta _{i}\\g_{pq}\varphi _{i}^{p}\varphi _{j}^{q}&=g_{ij}-\eta _{i}\eta _{j}.\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff8d09b0a4ae00ed6eb100b2d57bc118f275a0b)
That is to say that, relative to
the vector
has length one and is orthogonal to
furthermore the restriction of
to
is a Hermitian metric relative to the almost-complex structure
One says that
is an almost-contact metric manifold.
An almost-contact metric manifold
is said to be a Kenmotsu manifold if
![{\displaystyle \nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23b3a95a94312d92727531a2d4704effa7c6d794)