On the Splitting Tensor of the Weak f-Contact Structure

Abstract

A weak f-contact structure, introduced in our recent works, generalizes the classical f-contact structure on a smooth manifold, and its characteristic distribution defines a totally geodesic foliation with flat leaves. We find the splitting tensor of this foliation and use it to show positive definiteness of the Jacobi operators in the characteristic directions and to obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds, and prove integral formulas for a compact weak f-contact manifold. Based on applications of the weak f-contact structure in Riemannian contact geometry considered in the article, we expect that this structure will also be fruitful in theoretical physics, e.g., in QFT.

Characteristics of Sasakian Manifolds Admitting Almost ∗-Ricci Solitons

“…The article continues our study [24,25,28] of the geometry of weak f -contact manifolds. Following the approach in [23] for s = 1, we study the sectional and Ricci curvature in the ξdirections of a weak f -contact structure and its particular case -a weak f -K-contact structure.…”

“…We generalize this for a weak f -contact structure. Proposition 4.1 (see [24,25]). For a weak f -contact structure (f, Q, ξ i , η i , g), the tensor h i and its conjugate h * i satisfy g((…”

“…In [24,25,28] 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 skewsymmetric tensor) was initiated. This generalizes a metric f -structure and its satellites, and allows a new look at the classical theory and new applications.…”

“…In [24], we proved that the S-structure is rigid, i.e., our weak S-structure is the S-structure, and a weak metric f -structure with parallel tensor f is a weak C-structure. In [25], we obtained a topological obstruction (including the Adams number) to the existence of weak f -K-contact manifolds. In [28], we retracted weak structures with positive partial Ricci curvature onto the subspace of classical structures of the same type.…”