Homogeneous spaces, dynamics, cosmology: Geometric flows and rational dynamics
Abdelghani Zeghib (UMPA-ENSL)
TL;DR
This paper introduces Bianchi-Ricci and Bianchi-Einstein flows, which are simplified dynamical systems derived from Einstein's equations and Ricci flow for homogeneous spaces, providing insights into geometric evolution in cosmology.
Contribution
It offers an introductory analysis of Bianchi-Ricci and Bianchi-Einstein flows, connecting geometric flows with cosmological models through homogeneous metrics.
Findings
Bianchi-Ricci flows model geometric evolution in homogeneous spaces.
Bianchi-Einstein flows relate to cosmological solutions of Einstein's equations.
The approach simplifies complex PDEs to finite-dimensional dynamical systems.
Abstract
The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimensional dynamical systems, obtained by considering homogeneous metrics.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Advanced Differential Geometry Research