On the characterization of non-degenerate foliations of pseudo-Riemannian manifolds with conformally flat leaves

TL;DR

This paper establishes a necessary and sufficient tensor-based condition for the conformal flatness of leaves in non-degenerate foliations of pseudo-Riemannian manifolds, extending classical conformal flatness criteria.

Contribution

It introduces new tensor conditions involving the bi-conformal connection that characterize conformally flat leaves in pseudo-Riemannian foliations.

Findings

Provides a tensor-based criterion for conformal flatness of leaves.

Extends classical Weyl and Cotton tensor conditions to foliated pseudo-Riemannian manifolds.

Defines new tensors related to the bi-conformal connection for this purpose.

Abstract

A necessary and sufficient condition for the leaves of a {\em non-degenerate} foliation of a pseudo-Riemannian manifold to be conformally flat is developed. The condition mimics the classical condition of the vanishing of the Weyl or Cotton tensor establishing the conformal flatness of a pseudo-Riemannian manifold in the sense that it is also formulated in terms of the vanishing of certain tensors. These tensors play the role of the Weyl or the Cotton tensors and they are defined in terms of the the curvature of a linear torsion-free connection (the {\em bi-conformal connection}).