2 4 6 8 10 12 14 0.0 0.5 1.0 1.5 2.0 2.5 GLYPH<CapOmega> Δ (a) α ′ > 0 2 4 6 8 10 12 14 GLYPH<Minus> 2.5 GLYPH<Minus> 2.0 GLYPH<Minus> 1.5 GLYPH<Minus> 1.0 GLYPH<Minus> 0.5 0.0 GLYPH<CapOmega> Δ (b) α ′ < 0 FIG. 12: ¯ λ$_{s}$ vs ¯ Ω$_{0}$. The plots for ± ln ¯ Ω (continuous line) and δ at different orders, 2 nd (long dashed line), 3 rd (short dashed line) and 4 th (dotted line). V. SUMMARY AND CONCLUSION In this article we have investigated the higher order flows on various manifolds, from uniform homogeneous spaces to warped manifolds. Let us summarize the work done and mention some future possibilities briefly. We began with higher order geometric flows for the toy examples of the two sphere and hyperbolic space in two dimensions. It turns out that the results for the 1st order (Ricci) and 3rd order flows are similar in nature while the 2nd and 4th order flows have similarities. There is a marked difference for flows with α ′ > 0 and α ′ < 0, which has been pointed out. The existence of fixed points (solitons) have also been mentioned. We emphasize that these toy examples, though simplistic, do provide us with pointers towards distinguishing between the specific effects that occur because of the inclusion of higher order terms. As expected, the higher order terms do not resolve the lower order singularities though there are definite changes in terms of the singularity ‘time’, initial condition dependencies and the appearance of solitons. Further, we discuss the perturbative RG flow domain. Here, we are able to quantify the corrections due to the presence of the higher order terms. Subsequently, we turned towards higher order flows in the context of warped product manifolds. This analysis is largely inspired by the physics of warped extra dimensions and brane–bulk models. As claimed in Prop.IV.1 we show that the conformally AdS spacetime in Eq.(26) is a solution of the RG flow upto order N = 4 in α ′ . While reducing the RG flow PDEs to ODEs, assuming separability of the functions in the line element, we noted that the conformal AdS nature arises by using just the leading order terms. The higher order 22