On the Existence of a Codimension 1 Completely Integrable Totally Geodesic Distribution on a Pseudo-Riemannian Heisenberg Group

TL;DR

This paper demonstrates that the pseudo-Riemannian Heisenberg group uniquely admits a codimension 1 totally geodesic distribution that is completely integrable, contrasting with the Riemannian case where such distributions do not exist.

Contribution

It establishes the existence of a specific integrable geodesic distribution on the pseudo-Riemannian Heisenberg group, contrasting prior results in the Riemannian setting.

Findings

Existence of a codimension 1 integrable totally geodesic distribution on the pseudo-Riemannian Heisenberg group

Contrasts with the Riemannian case where such distributions do not exist

Provides new insights into the geometry of pseudo-Riemannian groups

Abstract

In this note we prove that the Heisenberg group with a left-invariant pseudo-Riemannian metric admits a completely integrable totally geodesic distribution of codimension 1. This is on the contrary to the Riemannian case, as it was proved by T. Hangan.