Riemann-Christoffel flows

TL;DR

This paper introduces a new geometric flow based on the Riemann-Christoffel curvature tensor, extending Ricci flow concepts to higher dimensions and Lorentzian manifolds, with applications to Einstein equations and cosmological models.

Contribution

It presents a novel curvature flow that incorporates all components of intrinsic curvature, generalizing Ricci flow to higher-dimensional and Lorentzian geometries.

Findings

Solutions to Einstein equations are examples of Riemann-Christoffel flows.

The flow applies to four-dimensional Lorentzian manifolds, including cosmological models.

Possible generalizations of the flow are discussed.

Abstract

A geometric flow based in the Riemann-Christoffel curvature tensor that in two dimensions has some common features with the usual Ricci flow is presented. For $n$ dimensional spaces this new flow takes into account all the components of the intrinsic curvature. For four dimensional Lorentzian manifolds it is found that the solutions of the Einstein equations associated to a "detonant" sphere of matter, as well, as a Friedman-Roberson-Walker cosmological model are examples of Riemann-Christoffel flows. Possible generalizations are mentioned.