Orthogonal foliations on riemannian manifolds

TL;DR

This paper derives an equation linking Ricci curvature and second fundamental forms of orthogonal foliations on Riemannian manifolds, providing conditions for local product structures and establishing an integral formula.

Contribution

It introduces a new equation relating Ricci curvature and second fundamental forms of orthogonal foliations, and offers conditions for local Riemannian product structures.

Findings

Derived an equation relating Ricci curvature and second fundamental forms.

Provided a sufficient condition for a manifold to be locally a Riemannian product.

Proved an integral formula for orthogonal foliations.

Abstract

In this work, we find an equation that relates the Ricci curvature of a riemannian manifold $M$ and the second fundamental forms of two orthogonal foliations of complementary dimensions, $\mathcal{F}$ and $\mathcal{F}^{\bot}$, defined on $M$. Using this equation, we show a sufficient condition for the manifold M to be locally a riemannian product of the leaves of $\mathcal{F}$ and $\mathcal{F}^{\bot}$, if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.