On the existence of geodesic vector fields on closed surfaces

TL;DR

This paper constructs a specific Riemannian metric on the 2-torus demonstrating that its universal cover lacks global Riemann normal coordinates, challenging assumptions about coordinate systems on closed surfaces.

Contribution

It provides a counterexample of a Riemannian metric on the 2-torus with unique geometric properties affecting coordinate existence.

Findings

Universal cover of the constructed metric does not admit global Riemann normal coordinates.

The example challenges previous beliefs about coordinate systems on closed surfaces.

The construction impacts understanding of geodesic vector fields on closed surfaces.

Abstract

We construct an example of a Riemannian metric on the 2-torus such that its universal cover does not admit global Riemann normal coordinates.