Locally Conformal C ₆-Manifolds and Generalized Sasakian-Space-Forms

Abstract

An algebraic characterization of generalized Sasakian-space-forms is stated. Then, one studies the almost contact metric manifolds which are locally conformal to C ₆-manifolds, simply called l.c. C ₆-manifolds. In dimension 2n + 1 ≥ 5, any of these manifolds turns out to be locally conformal cosymplectic or globally conformal to a Sasakian manifold. Curvature properties of l.c. C ₆-manifolds are obtained, with particular attention to the k-nullity condition. This allows one to state a local classification theorem, in dimension 2n + 1 ≥ 5, under the hypothesis of constant sectional curvature. Moreover, one proves that an l.c. C ₆–manifold is a generalized Sasakian-space-form if and only if it satisfies the k-nullity condition and has pointwise constant ${\varphi}$-sectional curvature. Finally, local classification theorems for the generalized Sasakian-space-forms in the considered class are obtained.