Let be an almost-contact manifold. One says that a Riemannian metric on is adapted to the almost-contact structure if:
That is to say that, relative to the vector has length one and is orthogonal to furthermore the restriction of to is a Hermitian metric relative to the almost-complex structure One says that is an almost-contact metric manifold.
An almost-contact metric manifold is said to be a Kenmotsu manifold if