Geometry of a weak para-$f$-structure

Abstract

We study geometry of the weak almost para-$f$-structure and its subclasses. This allow us to produce totally geodesic foliations and also to take a fresh look at the para-$f$-structure introduced by A.\,Bucki and A.\,Miernowski. We demonstrate this by generalizing several known results on almost para-$f$-manifolds. First, we express the covariant derivative of $f$ using a new tensor on a metric weak para-$f$-structure, then we prove that on a weak para-$\mathcal{K}$-manifold the characteristic vector fields are Killing and $\ker f$ defines a totally geodesic foliation. Next, we show that a para-$\mathcal{S}$-structure is rigid (i.e., a weak para-$\mathcal{S}$-structure is a para-$\mathcal{S}$-structure), and that a metric weak para-$f$-structure with parallel tensor $f$ reduces to a weak para-$\mathcal{C}$-structure. We obtain corollaries for $p=1$, i.e., for a weak almost paracontact structure.

13 pages. arXiv admin note: substantial text overlap with arXiv:2205.02158