Weak Nearly $\mathcal S$- and Weak Nearly $\mathcal C$- Manifolds
Vladimir Rovenski
TL;DR
The paper develops weak nearly ${\cal S}$- and weak nearly ${\cal C}$-structures on weak metric $f$-manifolds, extending classical nearly Kähler, Sasakian, and cosymplectic geometries. It defines these structures via the symmetric part of the covariant derivative of the $f$-tensor and investigates their geometric consequences, including Killing properties of the Reeb fields, flat foliations, and local product decompositions when Reeb fields are parallel. A substantial portion is devoted to submanifold theory: the induced weak metric $f$-structure on submanifolds of nearly Kähler manifolds inherits weak nearly ${\cal S}$- or ${\cal C}$-structure under natural curvature and second fundamental form constraints, with quasi-umbilical conditions playing a central role. The results connect to physical models such as non-symmetric gravitational theories and multi-time Hamiltonian systems, suggesting broad applicability of weak nearly $f$-structures in geometry and physics.
Abstract
The recent interest of geometers in the $f$-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric $f$-structures on a smooth manifold, recently introduced by the author and R. Wolak, open a new perspective on the theory of classical structures. In the paper, we define structures of this kind, called weak nearly ${\cal S}$- and weak nearly ${\cal C}$- structures, study their geometry, e.g. their relations to Killing vector fields, and characterize weak nearly ${\cal S}$- and weak nearly ${\cal S}$- submanifolds in a weak nearly Kähler manifold.