K-manifolds locally described by Sasaki manifolds

Abstract

K-manifolds are normal metric globally framed f -manifolds whose Sasaki 2-form is closed. We introduce and study some subclasses of K-manifolds. We describe some examples and we also state local decomposition theorems.

S-Manifolds Versus Indefinite S-Manifolds and Local Decomposition Theorems

“…One can easily extend such a notion to more general contexts. On a metric (g.f.f )-manifold (M 2n+s , φ, ξ i , η i , g), by a D a -homothetic deformation of constant a we mean the following change of the structure tensors, [8]:…”

“…Firstly we recall that an almost contact metric structure (φ, ξ, η, g) on a manifold M 2n+1 is called an α-Sasakian structure, α > 0, if it is normal and its Sasaki 2-form Φ verifies dη = αΦ. In [8] the authors have proved that an S-manifold (M 2n+s , φ, ξ i , η i , g) is locally a Riemannian product of a √ s -Sasakian manifold and an (s − 1)-dimensional flat manifold. We are going to discuss a similar problem in Lorentz context, remarking that the distributions considered by the authors in [8] have to be adapted to the Lorentz case because they are neither orthogonal nor parallel with respect to the Levi-Civita connection when the metric is Lorentz.…”

“…In [8] the authors have proved that an S-manifold (M 2n+s , φ, ξ i , η i , g) is locally a Riemannian product of a √ s -Sasakian manifold and an (s − 1)-dimensional flat manifold. We are going to discuss a similar problem in Lorentz context, remarking that the distributions considered by the authors in [8] have to be adapted to the Lorentz case because they are neither orthogonal nor parallel with respect to the Levi-Civita connection when the metric is Lorentz. Following [1] we recall some fundamental notion about distributions and foliations in semi-Riemannian manifolds.…”