0.0 0.5 1.0 1.5 0 1 2 3 4 5 (a)( A 0 , B 0 ) = (5 , 3) , α ′ > 0 , λ ′ s = 1 . 6041 0 2 4 6 8 10 5 10 15 20 25 (b)( A 0 , B 0 ) = (5 , 3) , α ′ < 0 GLYPH<Minus> 1.0 GLYPH<Minus> 0.5 0.0 0.5 1.0 0 2 4 6 8 (c)( A 0 , B 0 ) = (5 , 3) , α ′ > 0 GLYPH<Minus> 0.5 0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 (d)( A 0 , B 0 ) = (5 , 3) , α ′ < 0 FIG. 7: ( A ( λ ′ ), B ( λ ′ )) vs λ for 2 nd order flow ((a),(b))and 3 rd order flow((c),(d)) The constant terms are C$_{2}$ = ln( a ′ 2 0 − 1) − a ′ 2 0 and ˜ C$_{3}$ = C + 3 π 8 + ln 4 2 where C comes from Eq.14. The expansions for α ′ < 0 (spherical surface) and hyperbolic surface (both α ′ < 0 and α ′ > 0) can be obtained in a similar fashion though the α ′ < 0 case is not important in the σ -model context. Eq.24 can be written as − 2 λ ′ + C = a ′ $^{2}$+ δ where δ represents the corrections over the Ricci flow due to the higher order terms in the RG flow equation. We plot these δ at various order in Fig.8 and Fig. 9. It is clear that within the RG flow domain the curvature squared terms correspond to corrections which can be quantified order by order in perturbation theory. IV. HIGHER ORDER FLOW ON WARPED PRODUCT MANIFOLDS A. Geometric flow analysis We now consider the behaviour of a five dimensional warped product manifold under the flow Eq.(1). The metric on the manifold we consider can be written as, 14 ds 2 = e 2 f ( σ,λ $^{)}$η$_{µν}$ dx $^{µ}$dxν + r $^{2}$( σ, λ ) dσ 2 (25)