Geodesible contact structures on 3--manifolds

TL;DR

This paper classifies certain contact structures on closed 3-manifolds that are totally geodesic with respect to some Riemannian metric, focusing on Seifert manifolds and their twisting properties.

Contribution

It provides a complete classification of contact structures with negative twisting number on non-spherical Seifert manifolds, extending understanding of geodesible contact structures.

Findings

Classified contact structures with negative twisting on non-spherical Seifert manifolds.

Obtained partial results for spherical base cases.

Connected geodesibility with transverse contact structures.

Abstract

In this paper, we study and almost completely classify contact structures on closed 3--manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complete classification of contact structures with negative (maximal) twisting number (which includes the transverse ones) on Seifert manifolds whose base is not a sphere, as well as partial results in the spherical case.