Metric structures that admit totally geodesic foliations

Abstract

Motivated by the problem of finding suitable structures on a manifold M to obtain totally geodesic foliations, we recently introduced the weakened framed f-structure, i.e., the complex structure on f(TM) is replaced by a nonsingular skew-symmetric tensor, and its subclasses of weak ${{\mathcal {K}}}$-, ${{\mathcal {S}}}$-, and ${{\mathcal {C}}}$- structures. This allow us to take a fresh look at the classical f-structure by K. Yano, and subsequently studied by a number of geometers. 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 ${{\mathcal {K}}}$-manifold the characteristic vector fields are Killing and $\ker f$ defines a totally geodesic foliation, an ${{\mathcal {S}}}$-structure is rigid, i.e., our weak ${{\mathcal {S}}}$-structure is an ${{\mathcal {S}}}$-structure, and a metric weak f-structure with parallel tensor f reduces to a weak ${{\mathcal {C}}}$-structure. We obtain several corollaries for weak almost contact, weak cosymplectic and weak Sasakian structures.