Kenmotsu 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.

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.^[1]

An almost-contact metric manifold ${\displaystyle (M,\varphi ,\xi ,\eta ,g)}$ is said to be a Kenmotsu manifold if^[2] $${\displaystyle \nabla _{i}\varphi _{j}^{k}=-\eta _{j}\varphi _{i}^{k}-g_{ip}\varphi _{j}^{p}\xi ^{k}.}$$

Sources

• Blair, David E. (2010). Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics. Vol. 203 (Second edition of 2002 original ed.). Boston, MA: Birkhäuser Boston, Ltd. doi:10.1007/978-0-8176-4959-3. ISBN 978-0-8176-4958-6. MR 2682326. Zbl 1246.53001.
• Kenmotsu, Katsuei (1972). "A class of almost contact Riemannian manifolds". Tohoku Mathematical Journal. Second Series. 24 (1): 93–103. doi:10.2748/tmj/1178241594. MR 0319102. Zbl 0245.53040.

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