Kenmotsu manifold

Short description: 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 $(M,\phi ,\xi ,\eta )$ be an almost-contact manifold. One says that a Riemannian metric $g$ on $M$ is adapted to the almost-contact structure $(\phi ,\xi ,\eta )$ if: $$\begin{array}{rl}{g}_{ij}{\xi }^j& ={\eta }_i\\ g_{pq}{\phi }_i^p{\phi }_j^q& =g_{ij}-{\eta }_i{\eta }_j.\end{array}$$ 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 ${\phi }_p{|}_{\ker \left({\eta }_p\right)}.$ One says that $(M,\phi ,\xi ,\eta ,g)$ is an almost-contact metric manifold.({{{1}}}, {{{2}}})

An almost-contact metric manifold $(M,\phi ,\xi ,\eta ,g)$ is said to be a Kenmotsu manifold if({{{1}}}, {{{2}}}) $${\mathrm{\nabla }}_i{\phi }_j^k=-{\eta }_j{\phi }_i^k-g_{ip}{\phi }_j^p{\xi }^k.$$

References

Sources

• Blair, David E. (2010). Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics. 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. 
• Kenmotsu, Katsuei (1972). "A class of almost contact Riemannian manifolds". Tohoku Mathematical Journal. Second Series 24 (1): 93–103. doi:10.2748/tmj/1178241594.