Einstein metrics and Ricci solitons

introduction

A Riemannian manifold \((M, g)\) is called Einstein if it has constant Ricci curvature, i.e. \(Ric_g=\kappa \cdot g\) for some \(\kappa\in \mathbb{R}\)
In local coordinates,

\[ R_{ab} = \kappa\,g_{ab} \]

Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons
Ricci solitons on Finsler spaces are a generalization of Einstein spaces, which can be considered as a solution to the Ricci flow on compact Finsler manifolds.

\[R_{ab} - \frac{1}{2}g_{ab}R + g_{ab}\Lambda = 8\pi T_{ab}, \] written in geometrized units with G = c = 1.

The stress–energy tensor T_ab gives the matter and energy content of the underlying spacetime.
In a vacuum (a region of spacetime with no matter) T_ab = 0, and one can rewrite Einstein's equation in the form (assuming n > 2):

\[R_{ab} = \frac{2\Lambda}{n-2}\,g_{ab}.\]

Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.

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