Foliations transverse to a closed conformal vector field

TL;DR

This paper explores the geometric structure of codimension one foliations on Riemannian manifolds with closed conformal vector fields, focusing on conditions for totally geodesic leaves and characterizations of Montiel Foliations.

Contribution

It introduces the concept of Montiel Foliations and analyzes conditions under which foliations have special geometric properties like minimality and constant mean curvature.

Findings

Conditions for totally geodesic leaves in Montiel Foliations

Characterization of minimal and constant mean curvature foliations

Relation between totally geodesic foliations and Montiel Foliations

Abstract

In this article, we study the geometric properties of codimension one foliations on Riemannian manifolds equipped with vector fields that are closed and conformal. Apart from its singularities, these vector fields define codimension one foliations with nice geometric features, which we call Montiel Foliations. We investigate conditions for which a foliation transverse to this structure has totally geodesic leaves, as well as how the ambient space and the geometry of the leaves forces a given foliation into a Montiel Foliation. Our main results concern minimal leaves and constant mean curvature foliations, having compact or complete noncompact leaves. Finally, we characterize totally geodesic foliations by means of its relation to a prior Montiel Foliation.