7. Higher order flows on Ag S 2 can ⊕ Bg N can We now briefly discuss flows on an unwarped product manifold before moving on to warped products in the next section. Let N be a manifold(surface) of genus g ≥ 2 and g N can is a constant negative curvature( − 1). From Eqn.10 we can write the flow equations up to 4 th order (note that they are decoupled equations because the product is unwarped): In this case, for 2 nd and 4 th order flow we can have ancient as well as immortal solution depending on α ′ ( > 0 or < 0), respectively. Here Fig.7(a) and Fig.7(b) show such examples (for 2 nd order), of ancient and immortal solutions, respectively. We can also see from Fig.7(b) that the scale factor corresponding to S 2 approaches a constant for α ′ < 0. The 3 rd order flow is neither ancient nor immortal but maintains the same profile with the rest of its class. This may be seen in Fig.7(c) and Fig.7(d) for both choices of α ′ . Here the continuous and dashed lines represent the solutions for A ( λ ′ ) and B ( λ ′ ) respectively. B. Perturbative RG flow domain Finally, we look at the case where 1 x ′ 2 ≪ 1 or x ′ 2 ≫ 1. In this regime, the RG Flow assumption of small curvatures is valid. We look at the behavior of the flow equations(Eq.10 and Eq.11) at various orders, for spherical and hyperbolic surfaces. To obtain the correction at various orders, we expand the solutions at various orders and try to understand the characteristics of the RG flow within the abovementioned limit. To achieve our goals we consider a power series in 1 x ′ $_{2}$. Below, we show the expansion for α ′ > 0 in the case of a spherical surface at 2 nd and 3 rd order respectively. 13 dA dλ ′ = − 2 − α ′ 2 A − α ′ $_{2}$5 2 1 A 2 − 2 α ′ $^{3}$( 59 24 ζ (3) + 29 24 ) 1 A 3 (22) dB dλ ′ = 2 − α ′ 2 B + α ′ $_{2}$5 2 1 B 2 − 2 α ′ $^{3}$( 59 24 ζ (3) + 29 24 ) 1 B 3 (23) − 2 λ ′ + ˜ C$_{3}$ = a ′ 2 + ln( 1 a ′ $_{2}$) + 1 4 a ′ 2 − 3 4 a ′ 4 + 19 48 a ′ 6 + . . . (24b) − 2 λ ′ + C$_{2}$ = a ′ 2 + ln( 1 a ′ $_{2}$) − 1 a ′ 2 + 1 2 a ′ 4 − 1 3 a ′ 6 + . . . (24a)