On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

On the geometry of a weakened $f$-structure

“…For p = 1, from Theorem 4.1 we have the following Corollary 4.2 (see [11]). A weak almost contact metric structure on M 2n+1 is weak Sasakian if and only if it is a Sasakian structure (i.e., a normal weak contact metric structure) on M 2n+1 .…”

“…Then (3) holds and Theorem 5.2 can be applied. For p = 1, from Theorem 5.2 we have the following Corollary 5.1 (see [11]). Any weak almost contact structure (ϕ, Q, ξ, η, g) with the property ∇ϕ = 0 is a weak cosymplectic structure, i.e., dΦ = 0 and dη = 0, with vanishing tensor N (5) .…”

“…A natural question arises: How rich are metric weak f -structures compared to the classical ones? We study this question for weak K-, Sand C-structures; the results can be compared with corresponding results (p = 1) in [11] for weak almost contact, weak cosymplectic, and weak Sasakian structures. The question for a metric weak f -structure in general and further investigation of the curvature of metric f -manifolds and their connection with such popular structures as warped products and Ricci solitons are postponed to the future.…”

“…Proof. Using condition ∇f = 0, from (11) we obtain [f, f ] = 0. Hence, from (5) we get N (1) (X, Y ) = 2 i dη i (X, Y ) ξ i , and from (12) we obtain…”

Generalized Ricci solitons and Einstein metrics on weak $K$-contact manifolds

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We study metric structures on a smooth manifold (introduced in our recent works [8,10] and called a weak contact metric structure and a weak K-structure) which generalize the metric contact and K-contact structures, and allow a new look at the classical theory. First, we characterize weak K-contact manifolds among all weak contact metric manifolds by the property well known for K-contact manifolds, and find when a Riemannian manifold endowed with a unit Killing vector field forms a weak K-contact structure.

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