Projective Collineations of Decomposable Spacetimes Generated by the Lie Point Symmetries of Geodesic Equations

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 n+1-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.

PROPER PROJECTIVE SYMMETRY IN KANTOWSKI–SACHS AND BIANCHI TYPE III SPACETIMES

A study of Kantowski–Sachs and Bianchi type III spacetimes according to their proper projective symmetry is given by using the algebraic and direct integration techniques. It is shown that the above spacetimes do not admit proper projective collineations.

PROPER PROJECTIVE SYMMETRY IN THE SCHWARZSCHILD METRIC

A study of proper projective symmetry in the Schwarzschild metric is given by using algebric and direct integration techniques. It is shown that projective collineations admitted by the above metric are the Killing vector fields.