On the geometry of a weakened $f$-structure

Abstract

An f -structure, introduced by K. Yano in 1963 and subsequently studied by a number of geometers, is a higher dimensional analog of almost complex and almost contact structures. We recently introduced the weakened (globally framed) f -structure and its subclasses of weak K-, S-, and C-structures on Riemannian manifolds with totally geodesic foliations, which allow us to take a fresh look at the classical theory. We demonstrate this by generalizing several known results on framed f -manifolds. First, we express the covariant derivative of f using a new tensor on a metric weak f -structure, then we prove that on a weak K-manifold the characteristic vector fields are Killing and ker f defines a totally geodesic foliation, an S-structure is rigid, i.e., our weak S-structure is an S-structure, and a metric weak f -structure with parallel tensor f reduces to a weak C-structure.

On the Splitting Tensor of the Weak f-Contact Structure

“…The article continues our study [24,25] of the geometry of weak f -contact manifolds. Our achievement is the generalization of some results on f -contact manifolds to the case of weak f -contact manifolds and demonstration of the usefulness of this weak structure for the study of totally geodesic foliations, Killing vector fields and the corresponding splitting tensors on Riemannian manifolds.…”

“…In [25], we retracted weak structures with positive partial Ricci curvature onto the subspace of classical structures of the same type. In [24], we proved that the S-structure is rigid, i.e., our weak S-structure is the S-structure, and a metric weak f -structure with parallel tensor f is the weak C-structure. The characteristic distribution of the weak f -contact structure (and its particular case-the weak f -K-contact structure) is integrable and defines a totally geodesic foliation.…”

“…In [24,25] the study of "weak" contact structures on a smooth (2n + s)-dimensional manifold (i.e., the complex structure on the characteristic distribution is replaced with a nonsingular skew-symmetric tensor) was initiated. These structures generalize an f -contact structure and its satellites, and allow a new look at the Riemannian classical theory and find its new applications, for example, to build manifolds with Killing vector fields and totally geodesic foliations, (well known examples are flows on the sphere, linear flow on the torus, see Figure 1), to produce Riemannian manifolds with positive ξ-sectional curvature, and to investigate Einstein-type metrics.…”

Extrinsic Geometry of Foliations

“…Proof. Using condition ∇f = 0, from (10) we obtain [f, f ] = 0. Hence, from (4) we get N (1) (X, Y ) = −2 i dη i (X, Y ) ξ i , and from (11) we obtain…”

“…In [11], we introduced the "weak" metric structures that generalize an f -structure and a paraf -structure, and allow us to take a fresh look at the classical theory. In [10], we studied geometry of a weak f -structure and its satellites that are analogs of K-S-and C-manifolds. In this paper, using a similar approach, we study geometry of a weak para-f -structure and its important cases related to a pseudo-Riemannian manifold endowed with a totally geodesic foliation.…”