Kenmotsu manifold

Last updated August 12, 2023

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 $(M,\varphi ,\xi ,\eta )$ be an almost-contact manifold. One says that a Riemannian metric $g$ on $M$ is adapted to the almost-contact structure $(\varphi ,\xi ,\eta )$ if:

{\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 $g_{p},$ 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 $\varphi _{p}{\big \vert }_{\ker \left(\eta _{p}\right)}.$ One says that $(M,\varphi ,\xi ,\eta ,g)$ is an almost-contact metric manifold.^[1]

An almost-contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ is said to be a Kenmotsu manifold if^[2]

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

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References

↑ Blair 2010, p. 44.
↑ Blair 2010, p. 98.

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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[FOOTNOTEBlair201044-1] Blair 2010, p. 44.

[FOOTNOTEBlair201098-2] Blair 2010, p. 98.

[1]

[2]

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem