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In this work we present solutions to the Ricci flow equations in arbitrarydimensions, particularizing for the $3d$ and $4d$ cases. We start byconsidering the $3d$ case and note that our solutions belong to the family ofmaximally symmetric spaces that can be extended to the $d\geq 4$ case followingan analogue treatment. These solutions can be divided into two scenarios:maximally symmetric spaces with positive curvature i.e. de Sitter spaces, andmaximally symmetric spaces with negative curvature i.e. Anti-de Sitter spaces.We show that between both scenarios there is a {\it critical point} where thecurvature blows up along the flow. Also the solutions for $d\geq 4$ satisfy theflow equations with Riemannian or pseudo-Riemannian metrics due to the factthat the considered maximally symmetric spaces do not depend on time neither onthe angular coordinates yielding equations that depend only on the radialcoordinate and the Ricci flow parameter. Additionally we find an interestingeffect of the flow consisting in a change of the signature of the metric whenpassing the singular point. Besides the signature change, the sign of thecurvature also experiences a transition from positive to negative curvature,either with Riemannian or pseudo-Riemannian metrics throughout the singularpoint.

Evolution and metric signature change of maximally symmetric spaces under the Ricci flow