Weak Almost Contact Structures: a Survey

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.