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 $$(M, \varphi, \xi, \eta)$$ 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: $$\begin{align} g_{ij}\xi^j&=\eta_i\\ g_{pq}\varphi_i^p\varphi_j^q&=g_{ij}-\eta_i\eta_j. \end{align}$$ 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.

An almost-contact metric manifold $$(M, \varphi, \xi, \eta, g)$$ is said to be a Kenmotsu manifold if $$\nabla_i\varphi_j^k=-\eta_j\varphi_i^k-g_{ip}\varphi_j^p\xi^k.$$