On the curvature of a generalization of contact metric manifolds

Abstract

We consider a genaralization of contact metric manifolds given by assignment of 1-forms η 1 , . . . , η s and a compatible metric g on a manifold. With some integrability conditions they are called almost S-manifolds. We give a sufficient condition regarding the curvature of an almost S-manifold to be locally isometric to a product of a Euclidean space and a sphere.

Harmonic 1-Forms on Compact f-Manifolds

“…This result has already been obtained with a different proof in [7], in a more general context. If a 1 , . .…”

Scalar and φ-Sectional Curvature of a Certain Type of Metric f-Structures

“…If M is an S-manifold, then from (1.10) of [4] the left hand side of (3.6) is 0 and then ∇ϕ 2 = 4ns. The converse follows from Lemmas 3.3, 3.4 and Proposition 3.1 of [6].…”

“…The properties of this structure are studied in our previous paper (cf. [6]). However, the integrability properties of the structure Z 3 are fairly disappointing; for instance Z 3 can never be an almost S-manifold.…”

K-manifolds locally described by Sasaki manifolds

“…From [5] we know that a D a -homothetic deformation, a > 0, of the f.pkstructure (ϕ, ξ i , η i , g) on M 2n+s is a change of the structure tensors as follows: Now, we describe an example of K 0 -manifold which is not a C-manifold.…”

“…Such manifolds have been studied by several authors and from different point of view ( [1,3,4,5,7,12]). They are manifolds M 2n+s equipped with an f -structure ϕ of rank 2n with kernel parallelizable by s vector fields ξ 1 , .…”