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
David E. Blair. Riemannian geometry of contact and symplectic manifolds. Second edition. Progress in Mathematics, 203. Birkhäuser Boston, Ltd., Boston, MA, 2010. ISBN 978-0-8176-4958-6, doi:10.1007/978-0-8176-4959-3