Almost Contact Metric Structures on the Hypersurface of Almost Hermitian Manifolds

A note on W3-manifolds

“…As we have noted, an almost contact metric structure is induced on its oriented hypersurface N . The first group of the Cartan structural equations of such an almost contact metric structure looks as follows [12,17]:…”

“…If the hypersurface is totally umbilical, then its second fundamental form complies with the condition σ = λg, λ − const. Knowing how the matrix of the contravariant metric tensor on N looks [12]:…”

“…Let N be an oriented hypersurface in a Hermitian manifold M 2n and let σ be the second fundamental form of the immersion of N into M 2n . As it is wellknown [12,17], the almost Hermitian structure on M 2n induces an almost contact metric structure on N . We recall also that an almost contact metric structure on an odd-dimensional manifold N is defined by the system of tensor fields {Φ, ξ, η, g} on this manifold, where ξ is a vector field, η is a covector field, Φ is a tensor of the type (1, 1) and •, • is the Riemannian metric [13,15].…”

Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu type

“…. , n, and thus the Theorem 5.1 reduce to the following form: where C = (C i j ) and C −1 = ( C i j ) were defined in [5]. Then the substitution of the above relations in the second group of cartan structure equations, we conclude the following theorem: Theorem 5.5.…”

“…Now, we discuss the opposite problem, that is, if (N 2n , J, h) is an Hermitian manifold, then can we find a hypersuface of N which is the class of Kenmotsu type? We rely on the citation [5] for the background.…”