Evolution of Curvature in Riemannian Geometry

TL;DR

This paper investigates how the Riemann curvature tensor evolves under arbitrary timelike vector fields, demonstrating recurrence and divergence of geodesics, suggesting the existence of a non-singular, recurring cosmological model.

Contribution

It provides a novel analysis of curvature evolution in Riemannian geometry, linking it to Poincare's recurrence theorem and geodesic behavior in cosmology.

Findings

Curvature tensor evolution can be expressed via an arbitrary timelike vector field.

Poincare's recurrence theorem applies in the considered cosmological scenario.

Geodesics can both diverge and converge, indicating non-singular points.

Abstract

In this paper shall we endeavour to substantiate that the evolution of the Riemann- Christoffel tensor or curvature tensor can be expressed entirely by an arbitrary timelike vector field and that the curvature tensor returns to its initial value with respect to change in a particular index. This implies that Poincare's recurrence theorem is valid in this cosmological scenario. Also, it has been shown that geodesics can diverge just as they can converge. As is ostensible, this result indicates the existence the of a point of exclusivity - the opposite of a singularity.