On Almost-Riemannian Surfaces

TL;DR

This paper introduces almost-Riemannian surfaces, exploring their unique geometric and topological properties, especially where the structure degenerates, and highlights differences from classical Riemannian geometry.

Contribution

It provides a concise overview of 2D almost-Riemannian geometry, emphasizing its novel features and analyzing local and global properties of these surfaces.

Findings

Distribution is rank 2 almost everywhere but drops to rank 1 on a smooth curve.

Highlights differences between almost-Riemannian and classical Riemannian geometry.

Investigates topological, metric, and geometric aspects of almost-Riemannian surfaces.

Abstract

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its novelties with respect to Riemannian geometry. We present some results that investigate topological, metric and geometric aspects of almost- Riemannian surfaces from a local and global point of view.