Kenmotsu_manifold Knowpia

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 edit

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}.

References edit

^ 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.

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

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

[1]

[2]

Summary

Definitions edit

References edit