  Holographic RG flows dual to QFTs on maximally symmetric curved manifolds
(dS$_d$, AdS$_d$, and $S^d$) are considered in the framework of
Einstein-dilaton gravity in $d+1$ dimensions. A general dilaton potential is
used and the flows are driven by a scalar relevant operator. The general
properties of such flows are analyzed and the UV and IR asymptotics computed.
New RG flows can appear at finite curvature which do not have a zero curvature
counterpart. The so-called 'bouncing flows', where the $\beta$-function has a
branch cut at which it changes sign, are found to persist at finite curvature.
Novel quantum first-order phase transitions are found, triggered by a variation
in the $d$-dimensional curvature in theories allowing multiple ground states.
