Cosymplectic p-spheres

TL;DR

This paper introduces cosymplectic circles and spheres, classifies 3-manifolds admitting them, and explores their properties like tautness and roundness, establishing connections with complex and symplectic structures.

Contribution

It defines and classifies cosymplectic p-spheres, analyzes their properties, and links tautness and roundness to complex and conformal symplectic structures.

Findings

Complete classification of 3-manifolds with cosymplectic circles

Equivalence of roundness and tautness in dimension three

Examples of higher-dimensional cosymplectic circles with varied properties

Abstract

We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic $p$-sphere are studied. To any taut cosymplectic circle on a three-dimensional manifold $M$ we are able to canonically associate a complex structure and a conformal symplectic couple on $M \times \mathbb{R}$. We prove that a cosymplectic circle in dimension three is round if and only if it is taut. On the other hand, we provide examples in higher dimensions of cosymplectic circles which are taut but not round and examples of cosymplectic circles which are round but not taut.