On contact tops and integrable tops

TL;DR

This paper introduces a geometric structure called top, classifies manifolds admitting them, and describes their associated metrics, enriching the understanding of contact and integrable forms in Riemannian geometry.

Contribution

It defines the concept of tops as bundles over Riemannian 3-manifolds and classifies the manifolds that admit such structures, detailing their metrics.

Findings

Classification of manifolds admitting tops

Description of metrics associated with tops

Connection between contact/integrable forms and geometric structures

Abstract

In this paper, we introduce a geometric structure called top, which is a trivialized bundle of plane pencils over a Riemannian 3-manifold, defined as the set of kernels of a circle of 1-forms (e.g. of contact and integrable forms) with particular properties with respect to the metric. We classify the manifolds which admit tops and we describe the associated metrics.