Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator

We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn (c) satisfying the two conditions φ l = l φ , $\varphi l=l\varphi ,$ where l = R ( ⋅ , ξ ) ξ and φ $l=R(\cdot ,\xi )\xi \text{ and }\varphi $ is the almost contact metric structure of M, and ( ∇ ξ l ) X = $\left( {{\nabla }_{\xi }}l \right)X=$ ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.

Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers

We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type C5⊕C6⊕C12 of Chinea-González classification.

Lifting semi-invariant submanifolds to distribu­tion of almost contact metric manifolds

Let M be an almost contact metric manifold of dimension n = 2m + 1. The distribution D of the manifold M admits a natural structure of a smooth manifold of dimension n = 4m + 1. On the manifold M, is defined a linear connection that preserves the distribution D; this connection is determined by the interior connection that allows parallel transport of admissible vectors along admissible curves. The assigment of the linear connection is equivalent to the assignment of a Riemannian metric of the Sasaki type on the distribution D. Certain tensor field of type (1,1) on D defines a so-called prolonged almost contact metric structure. Each section of the distribution D defines a morphism of smooth manifolds. It is proved that if a semi-invariant sub­manifold of the manifold M and is a covariantly constant vec­tor field with respect to the N-connection , then is a semi-invariant submanifold of the manifold D with respect to the prolonged almost contact metric structure.

On a property of W4-manifolds

The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic. Moreover, the quasi-Sasakian structure is cosymplectic if and only if the hypersurface is a to­tally geodesic submanifold of the considered W4-manifold. From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally confor­mal Kählerian (LCK-) manifold also is either homothetic to a Sasakian structure or cosymplectic.

On nonexistence of Kenmotsu structure on аст-hypersurfaces of cosymplectic type of a Kählerian manifold

Almost contact metric (аст-)structures induced on oriented hypersurfaces of a Kählerian manifold are considered in the case when these аст- structures are of cosymplectic type, i. e. the contact form of these structures is closed. As it is known, the Kenmotsu structure is the most important non-trivial example of an almost contact metric structure of cosymplectic type. The Cartan structural equations of the almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold are obtained. It is proved that an almost contact metric structure of cosymplectic type on a hypersurface of a Kählerian manifold of dimension at least six cannot be a Kenmotsu structure. Moreover, it follows that oriented hypersurfaces of a Kählerian manifold of dimension at least six do not admit non-trivial almost contact metric structures of cosymplectic type that belong to any well studied class of аст-structures. The present results generalize some results on almost contact metric structures on hypersurfaces of an almost Hermitian manifold obtained earlier by V. F. Kirichenko, L. V. Stepanova, A. Abu-Saleem, M. B. Banaru and others.

A dimensional restriction for a class of contact manifolds

In this work we consider a class of contact manifolds (M, η) with an associated almost contact metric Structure (ϕ, ξ, η, g). This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class C9 ⊕ C10 defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension 4n + 1. Under the assumption that the sectional curvature of the horizontal 2-planes is constant at one point, we obtain that these manifolds must have dimension 5.

On a type of K-contact Riemannian manifold

Let M be an n-dimensional (n = 2m + 1, m ≦ 1) real differentiable manifold. if on M there exist a tensor field , a contravariant vector field ξi and a convariant vector field ηi such that then M is said to have an almost contact structure with the structure tensors (φ,ξ, η) [1], [2]. Further, if a positive definite Riemannian metric g satisfies the conditions then g is called an associated Riemannian metric to the almost contact structure and M is then said to have an almost contact metric structure. On the other hand, M is said to have a contact structure [2], [4] if there exists a 1-form η over M such that η ∧ (dη)m ≠ 0 everywhere over M where dη means the exterior derivation of η and the symbol ∧ means the exterior multiplication. In this case M is said to be a contact manifold with contact form η. It is known [2, Th. 3,1] that if η = ηidxi is a 1-form defining a contact structure, then there exists a positive definite Riemannian metric in gij such that and define an almost contact metric structure with and ηi where the symbol ∂i standing for ∂/∂xi.