I. INTRODUCTION AND OVERVIEW In the 1980s, Friedan [1] first analysed the renormalisation group (henceforth referred as RG) flows for nonlinear σ -models and showed how one could arrive at the vacuum Einstein equations R$_{ij}$ = 0 if one assumed quantum conformal (Weyl) invariance, thereby equating the metric β -functions to zero. This became a crucial result in the context of string the- ory and a route towards connecting/obtaining Einstein’s general relativity with/from string theory. Several authors, subsequently proved important generalisations [2] (particularly in the context of heterotic string theory), which provided further evidence of the abovemen- tioned connection. The fact that general relativity (modulo some additions) does emerge from string theory from the requirement of quantum conformal invariance, is a more or less accepted fact, today. Around the same time in the 1980s, Hamilton [3], in the context of mathematics, pro- posed the Ricci flow equations, with a largely different motivation, primarily related to the geometrisation conjecture of Thurston, which involves the topological classification of three manifolds. More recently, Perelman’s work [4] on Ricci flows and its utility in proving the Poincare conjecture has drawn many researchers to revisit such geometric flows[5]. It is well–known that the Ricci flow equations are the same (modulo a proper scaling of the flow parameter) as the first order RG flow equations of the nonlinear σ –model. However, one needs to keep in mind the fact that the RG flow equations are obtained from the perturbative evaluation of the β –function and thus, they approximate the full, non–perturbative (and as yet unknown) flow in the small curvature regime (i.e. α ′ R 2 c << 1, where 1 R 2 c is related to the curvature scale) [6]. On the other hand, from a different perspective, the same equations can be thought of as geometric flows where higher order terms are consistently included. We shall perform our analysis from both these points of view, in this article. Recently, various aspects (eg. irreversibility, gradient flow) of σ -model RG flows (with and without the dilaton and Kalb-Ramond field), at the lowest order have been investigated from a mathematical/geometrical perspective [7]. The connection between RG and geometric flows have been nicely discussed in [8], [9]. In particular, Carfora [8]makes a valid point about the connection through the following statement in his article: ‘the structure of Ricci flow singularities suggests a natural way for extending, beyond the weak coupling regime, the embedding of the Ricci flow into the renormalization group flow’ . 2