Weak Almost Contact Structures: a Survey

TL;DR

This survey explores weak almost contact structures, a generalization of contact manifolds, highlighting recent advances in their geometric properties and applications in Riemannian geometry.

Contribution

It provides a comprehensive overview of recent developments in weak almost contact structures, a novel generalization in contact geometry.

Findings

Analysis of geodesic and Killing fields in weak structures

Results on rigidity and splitting theorems

Insights into Ricci-type solitons and Einstein-type metrics

Abstract

Weak almost contact manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact manifolds. The paper surveys recent results (concerning geodesic and Killing fields, rigidity and splitting theorems, Ricci-type solitons and Einstein-type metrics, etc.) in this new field of Riemannian geometry.