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 K\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {K}}}$$\end{document}-, S\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {S}}}$$\end{document}-, and C\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {C}}}$$\end{document}- 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 K\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {K}}}$$\end{document}-manifold the characteristic vector fields are Killing and kerf\documentclass[12pt]{minimal}
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\begin{document}$$\ker f$$\end{document} defines a totally geodesic foliation, an S\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {S}}}$$\end{document}-structure is rigid, i.e., our weak S\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {S}}}$$\end{document}-structure is an S\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {S}}}$$\end{document}-structure, and a metric weak f-structure with parallel tensor f reduces to a weak C\documentclass[12pt]{minimal}
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\begin{document}$${{\mathcal {C}}}$$\end{document}-structure. We obtain several corollaries for weak almost contact, weak cosymplectic and weak Sasakian structures.
