Projective collineations of decomposable spacetimes generated by the Lie point symmetries of geodesic equations

TL;DR

This paper explores how Lie point symmetries of geodesic equations relate to projective collineations in decomposable spacetimes, revealing that these symmetries correspond to collineations of subspaces and demonstrating their applications.

Contribution

It establishes that projective collineations of decomposable spacetimes are equivalent to Lie point symmetries of geodesic equations in subspaces, providing a geometric construction approach.

Findings

Lie point symmetries correspond to projective collineations in decomposable spacetimes

The symmetries of geodesic equations are linked to collineations of subspaces

Application examples illustrate the theoretical results

Abstract

We investigate the relation of the Lie point symmetries for the geodesic equations with the collineations of decomposable spacetimes. We review previous results in the literature on the Lie point symmetries of the geodesic equations and we follow a previous proposed geometric construction approach for the symmetries of differential equations. In this study we prove that the projective collineations of a $\left( n+1\right) \,$% -dimensional decomposable Riemannian space are the Lie point symmetries for geodesic equations of the $n$-dimensional subspace. We demonstrate the application of our results with the presentation of applications.