On the geometry of a weakened $f$-structure

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, defined by a (1,1)-tensor field $f$ on a $(2n+p)$-dimensional manifold, which satisfies $f^3 + f = 0$ and has constant rank $2n$. We recently introduced the weakened (globally framed) $f$-structure (i.e., the complex structure on $f(TM)$ is replaced by a nonsingular skew-symmetric tensor) and its subclasses of weak $K$-, ${\cal S}$-, and ${\cal C}$- 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 globally 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 ${\cal S}$-structure is rigid, i.e., our weak ${\cal S}$-structure is an ${\cal S}$-structure, and a metric weak $f$-structure with parallel tensor $f$ reduces to a weak ${\cal C}$-structure. For $p=1$ we obtain the corresponding corollaries for weak almost contact, weak cosymplectic, and weak Sasakian structures.