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.
Definitions
Let [math]\displaystyle{ (M, \varphi, \xi, \eta) }[/math] be an almost-contact manifold. One says that a Riemannian metric [math]\displaystyle{ g }[/math] on [math]\displaystyle{ M }[/math] is adapted to the almost-contact structure [math]\displaystyle{ (\varphi, \xi, \eta) }[/math] if: [math]\displaystyle{ \begin{align} g_{ij}\xi^j&=\eta_i\\ g_{pq}\varphi_i^p\varphi_j^q&=g_{ij}-\eta_i\eta_j. \end{align} }[/math] That is to say that, relative to [math]\displaystyle{ g_p, }[/math] the vector [math]\displaystyle{ \xi_p }[/math] has length one and is orthogonal to [math]\displaystyle{ \ker \left(\eta_p\right); }[/math] furthermore the restriction of [math]\displaystyle{ g_p }[/math] to [math]\displaystyle{ \ker \left(\eta_p\right) }[/math]is a Hermitian metric relative to the almost-complex structure [math]\displaystyle{ \varphi_p\big\vert_{\ker \left(\eta_p\right)}. }[/math] One says that [math]\displaystyle{ (M, \varphi, \xi, \eta, g) }[/math] is an almost-contact metric manifold.({{{1}}}, {{{2}}})
An almost-contact metric manifold [math]\displaystyle{ (M, \varphi, \xi, \eta, g) }[/math] is said to be a Kenmotsu manifold if({{{1}}}, {{{2}}}) [math]\displaystyle{ \nabla_i\varphi_j^k=-\eta_j\varphi_i^k-g_{ip}\varphi_j^p\xi^k. }[/math]
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.
 |  |
