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.
, φ , ξ , η )
be an almost-contact manifold.
One says that a Riemannian metric
is adapted to the almost-contact structure
( φ , ξ , η )
{\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}}}
That is to say that, relative to
has length one and is orthogonal to
ker
ker
is a Hermitian metric relative to the almost-complex structure
ker
, φ , ξ , η , g )
is an almost-contact metric manifold.
[1] An almost-contact metric manifold
, φ , ξ , η , g )
is said to be a Kenmotsu manifold if[2]
{\displaystyle \nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.}
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