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