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.
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