Contact metric manifolds satisfying a nullity condition

C-totally real warped product submanifolds

“…Moreover, for κ = 1, the subbundle D = ker(η) can be decomposed in the eigenspace distributions D + and D − relative to the eigenvalues λ and −λ, respectively. These distributions are orthogonal to each other and have dimension n. There are many motivations for studying (κ, µ)-manifolds: the first is that, in the non-Sasakian case the condition (3.6) determines the curvature completely; moreover, while the values of κ and µ change, the form of (3.6) is invariant under D-homothetic deformations ( [3]); finally, there is a complete classification of these manifolds, given in [5] by E. Boeckx, who proved also that any non-Sasakian (κ, µ)-manifold is locally homogeneous and strongly locally ϕ-symmetric ( [4], [6]). There are also non-trivial examples of (κ, µ)-manifolds, the most important being the unit tangent sphere bundle T 1M of a Riemannian manifoldM of constant sectional curvature with the usual contact metric structure.…”

Generalized Sasakian-space-forms

“…Then, by virtue of Theorem 1 of [4] we can obtain the following results: And, what can we say about 3-dimensional contact metric generalized Sasakianspace-forms? First, let us mention that the writing of the curvature tensor of a 3-dimensional generalized Sasakian-space-form is not unique.…”

On N(k)-Contact Metric Manifolds

“…The contact metric manifold with ξ belonging to the k-nullity distribution is called N (k)-contact metric manifold and such a manifold is also studied by various authors. Generalizing this notion in 1995, Blair, Koufogiorgos and Papantoniou [4] introduced the notion of a contact metric manifold with ξ belonging to the (k, µ)-nullity distribution, where k and µ are real constants. In particular, if µ = 0, then the notion of (k, µ)-nullity distribution reduces to the notion of k-nullity distribution.…”