Geometry of weak metric $f$-manifolds: a survey

TL;DR

This survey reviews the geometry of weak metric $f$-manifolds, a generalization of classical $f$-structures, highlighting new applications in geometric analysis and differential geometry.

Contribution

It provides a comprehensive overview of weak metric $f$-manifolds, including their properties, classifications, and applications, extending classical $f$-structure theory.

Findings

Revisiting classical theory with weak $f$-structures

Discovering new applications in geometry

Classifying distinguished subclasses of weak metric $f$-manifolds

Abstract

A weak $f$-structure on a smooth manifold, introduced by the author and R. Wolak (2022), generalizes K. Yano's (1961) $f$-structure. This generalization allows us to revisit classical theory and discover new applications related to Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results on weak metric $f$-manifolds, where the complex structure on the contact distribution of a metric $f$-structure is replaced with a nonsingular skew-symmetric tensor, and explores its distinguished classes.