Conformal Geodesics Cannot Spiral

TL;DR

This paper proves that conformal geodesics on Riemannian manifolds cannot spiral, meaning they cannot be trapped in arbitrarily small neighborhoods of a point, which clarifies their geometric behavior.

Contribution

The paper establishes a fundamental property of conformal geodesics, demonstrating they cannot exhibit spiraling behavior on Riemannian manifolds.

Findings

Conformal geodesics cannot spiral or be trapped in neighborhoods.

The result clarifies the global behavior of conformal geodesics.

Provides a geometric constraint on conformal geodesic trajectories.

Abstract

We show that conformal geodesics on a Riemannian manifold cannot spiral: there does not exist a conformal geodesic which becomes trapped in every neighbourhood of a point.