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Kenmotsu manifold

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 ${\displaystyle (M,\varphi ,\xi ,\eta )}$ be an almost-contact manifold. One says that a Riemannian metric ${\displaystyle g}$ on ${\displaystyle M}$ is adapted to the almost-contact structure ${\displaystyle (\varphi ,\xi ,\eta )}$ 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}}}
That is to say that, relative to ${\displaystyle g_{p},}$ the vector ${\displaystyle \xi _{p}}$ has length one and is orthogonal to ${\displaystyle \ker \left(\eta _{p}\right);}$ furthermore the restriction of ${\displaystyle g_{p}}$ to ${\displaystyle \ker \left(\eta _{p}\right)}$is a Hermitian metric relative to the almost-complex structure ${\displaystyle \varphi _{p}{\big \vert }_{\ker \left(\eta _{p}\right)}.}$ One says that ${\displaystyle (M,\varphi ,\xi ,\eta ,g)}$ is an almost-contact metric manifold.

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

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

References

• David E. Blair. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010. ISBN 978-0-8176-4958-6, doi:10.1007/978-0-8176-4959-3
• Katsuei Kenmotsu. A class of almost contact Riemannian manifolds. Tohoku Mathematical Journal (2) 24 (1972), 93–103. doi:10.2748/tmj/1178241594