Geodesic rigidity of Levi-Civita connections admitting essential projective vector fields

TL;DR

This paper proves that certain 3D Riemannian and semi-Riemannian manifolds with specific projective vector fields are necessarily projectively flat, advancing understanding of geometric structures and symmetries.

Contribution

It establishes a new rigidity result linking the existence of non-linearizable singularities in projective vector fields to projective flatness in manifolds.

Findings

Manifolds with non-linearizable singularities in projective vector fields are projectively flat.

The result applies to connected 3D Riemannian and closed semi-Riemannian manifolds.

Provides conditions under which projective vector fields imply flatness.

Abstract

In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold $M^n$($n>1$) admitting a projective vector field with a non-linearizable singularity is projectively flat.