How to construct all metric f-K-contact manifolds

Abstract

We show that any compact metric f-K-contact, respectively S-manifold is obtained from a compact K-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.

Contact foliations and generalised Weinstein conjectures

“…Remark 2.35. Theorem 2.32 is somewhat similar in taste to the characterisation of metric f -K-structures provided by Goertsches and Loiudice in [24,Theorem 4.4]. Note that a metric f -K-structures induces a uniform q-contact structure for which all the Reeb fields are Killing for some metric.…”

“…It is also our first result specifically concerning contact foliations, in that it does not apply for contact flows, i.e., for the classic case. It is a result nicely related to a Theorem of Goertsches and Loiudice [24] stating that every metric f -K-contact manifold can be constructed from a K-contact manifold utilising mapping tori and the so-called "type II" deformations (cf. [24,Theorem 4.4]).…”

On the Splitting Tensor of the Weak f-Contact Structure

“…Is a compact weak f -K-contact Einstein manifold an S-manifold? When is a given weak f -K-contact manifold a mapping torus (see [17]) of a manifold of lower dimension? When does a weak f -contact manifold equipped with a Ricci-type soliton structure carry a canonical (for example, with constant sectional curvature or Einstein-type) metric?…”

“…The Jacobi operator R ξ (ξ ∈ ker f , ξ = 1) is defined as R ξ : X → R X, ξ ξ, e.g., [21]. We generalize the property of an f -K-contact manifold that the ξ-sectional curvature is constant and equal to 1, or, equivalently, R ξ i (X) = X (X ∈ D), see [17]. Again, the proof requires some calculations with the tensor N (5) .…”

“…The f -K-contact structure (i.e., an f -contact structure, whose characteristic vector fields generate 1-parameter groups of isometries), see [16], generalizes the K-contact structure of [3] (i.e., s = 1), both structures can be regarded as intermediate between a framed f -structure and S-structure (Sasaki structure when s = 1). In [17], conditions are found under which a given compact f -K-contact manifold is a mapping torus of such a manifold of lower dimension. Various symmetries of contact and framed f -manifolds are studied, e.g., in [18], and sufficient conditions are considered when an f -contact manifold carries a canonical metric, such as Einstein-type or constant curvature, or admits a local decomposition (splitting), in [4,19,20].…”