Second Order Renormalization Group Flow on Warped Product Manifolds

Çankaya University Journal of Science and Engineering Volume 16, No. 1 (2019) 081–097 Date Received: Oct 5, 2018 Date Accepted: Apr 29, 2019 Second Order Renormalization Group Flow on Warped Product Manifolds Zoheir Toeiserkani1 , Asadollah Razavi2 1 ,2 Department of Pure Mathematics, Faculity of Mthematics and Computer Sciences, Shahid Bhonar University of Kerman, Iran, e-mail: , Abstract: In this work we have studied the evolution of a warped product (WP) manifold under second order renormalization group (RG-2) flow. We have shown some conditions for the existence of a solution of RG-2 flow on WP manifolds. Also, we have found a necessary condition for warped function under RG-2 flow. In particular, we study some special WP metric of real line with a manifold. Eventually, by extending conditions to pseudo-Riemannian manifold, we find a PDE for Robertson-Walker (RW) metrics, and show that there is no RG-2 flow for RW metrics. Keywords: Ricci flow, second order renormalization group flow, warped product manifold. 1. Introduction The Ricci flow was introduced and studied by Hamilton [8], and has been a topic of interest in both mathematics and physics. The Ricci flow is an evolution equation for Riemannian metrics. In the Ricci flow, one begins with a smooth Riemannian manifold M, equipped with a smooth Riemannian metric g0 and evolves its metric by the equation ∂ ∂t g(t) = −2Ric(g(t)), g(0) = g0 , where t ∈ I, I is an interval, and Ric(g(t)) denotes the Ricci curvature of g(t). Many authors, have tried to extend Ricci flow from different point of views. The Ricci flow is the first-order approximation of renormalization group flow for nonlinear sigma models in quantum field theory. The second order approximation of the renormalization group flow for the nonlinear sigma model of quantum field theory, which we label by RG-2 flow, is specified by α ∂ g = −2Ric − Rm2 , ∂t 2 (1) Rm2i j = g pk gql gnm Riklm R j pqn , (2) where ISSN 1309 - 6788 c 2019 Çankaya University CUJSE 16, No. 1 (2019) Second Order Renormalization Group Flow on Warped Product Manifolds 81 denotes the quadratic curvature and α is a positive parameter. Note that for our purposes here, we can assume α to be non-negative. For α = 0, the system (1) reduces to the Ricci flow. RG-2 flow is diffeomorphism invariant but unlike the Ricci flow, it is not a weakly parabolic system. Gimre, Guenther and Isenberg, modifying De-Turck method for RG-2, proved conditions for the short time existence of the second order renormalization group flow in general dimension [4]. Some of the mathematical features of this flow have been studied in recent years, [5], [6], [7] and [14]. The concept of warped product (WP) metrics was first introduced by Bishop and O’Neill [1]. In Riemannian geometry, warped product manifolds have been used to construct new examples. The warped product B ×u F of two Riemannian manifolds B and F and real warped function u : B → R, is the product manifold B × F furnished with the metric g = gB + u2 gF . The Ricci flow on warped product manifolds was studied over the last few years [3], [12], [13], [16] and [17]. In this work, we have studied the property of second order renormalization group flow on warped product manifolds. First, we investigate an existence condition of RG-2 on WP manifolds, and extend the curvature criterion of short-time existence. Then, using WP metric curvature and RG-2 flow, we find some relations for warped function. Many exact solutions of the Einstein field equations and modified field equations are warped products, for instance, the Robertson-Walker (RW) models are warped products. Robertson and Walker independently showed in the mid-1930s that this is the most general metric possible for describing an expanding, homogeneous and isotropic universe. Hesamifard and Rezaii, studied RG-2 flow on RW metric in spherical coordinates [10]. Using property of RG-2 flow on WP manifolds, we have found a PDE, and we have studied some properties of its solution. We have shown as in [10], that there is no solution of RG-2 flow on RW manifolds. 2. Preliminaries For any closed Riemannian manifold (M, g0 ), and for all sectional curvatures KP (g0 ), at all point p ∈ M and planes P ⊂ Tp M, if 1 + αKP > 0, then there exists a unique solution g(t) of the initial value problem ∂t g = −Ric − α2 Rm2 , g(0) = g0 , on some time interval [0, T ) [4]. Let gFk be a Riemannian metric on an n-dimensional manifold F with constant curvature k, then RicF = k(n − 1)gFk and Rm2F = 2k2 (n − 1)gFk . If gF (t) is a solution of the second order renormalization group flow (1), with initial metric gF (0) = gFk , then gF (t) preserves its conformal class, 82 Z. Toeiserkani and A. Razavi and we may write gF (t) = φ (t)gFk , where φ (t) obtained by the following implicit function [5], [6] φ (t) = −2k(n − 1)t + 1 + αk 2φ (t) + αk Ln . 2 2 + αk Note that Ric(φ g) = Ric(g) and Rm2 (φ g) = φ1 Rm2 (g)). Let (B, gB ) and (F, gF ) be two (Pseudo-) Riemannian manifolds with dimensions m and n, respectively. Let M = B ×u F and g = gB + u2 gF where u : B → R is smooth positive function. For any point (x, y) ∈ M, and vectors X B ,Y B , Z B ... ∈ Tx B and X F ,Y F , Z F ... ∈ Ty F, we have for the Riemannian curvature of warped product manifold (M, g) [15]; R(X B ,Y B )Z F = R(X F ,Y F )Z B = 0, R(X B ,Y B )Z B = RB (X B ,Y B )Z B , R(X F ,Y B )Z B = 1u HessB (u)(Y B , Z B )X F , (3) R(X B ,Y F )Z F = ugF (Y F , Z F )∇BX B (∇B u), R(X F ,Y F )Z F = RF (X F ,Y F )Z F − |∇B u|2gB (gF (X F , Z F )Y F − gF (Y F , Z F )X F ). where, R, RB and RF are Riemannian curvatures of (M, g), (B, gB ) and (F, gF ), respectively. Also, we have for the Ricci tensor [15] Ric(X B .Y F ) = 0, Ric(X B ,Y B ) = RicB (X B ,Y B ) − nu HessB (u)(X B ,Y B ), (4) Ric(X F ,Y F ) = RicF (X F ,Y F ) − (u∆gB u + (n − 1)|∇B u|2gB )gF (X F ,Y F ), where, Ric, RicB and RicF define Ricci tensors of (M, g), (B, gB ) and (gF , F), respectively. Generaly, at a point (x, y) ∈ M and vectors X,Y ∈ T(x,y) M, where X = X B + X F , Y = Y B + Y F , X B ,Y B ∈ Tx B and X F ,Y F ∈ Ty F, we have Ric(X,Y ) = RicB (X B ,Y B ) + RicF (X F ,Y F ) − nu HessB (u)(X B ,Y B ) −u∆gB ugF (X F ,Y F ) − (n − 1)|∇B u|2gB gF (X F ,Y F ). Directly, we can calculate, the Riemannian curvature R(X,Y, Z,W ) = g(R(X,Y )Z,W ), and have: R(X B ,Y B , Z F ,W B ) = R(X F ,Y F , Z B ,W F ) = R(X B ,Y B , Z F ,W F ) = 0, R(X B ,Y B , Z B ,W B ) = RB (X B ,Y B , Z B ,W B ), R(X B ,Y F , Z B ,W F ) = −uHessB (u)(X B , Z B )gF (Y F ,W F ), R(X F ,Y F , Z F ,W F ) = (5) u2 RF (X F ,Y F , Z F ,W F )   −u2 |∇B u|2gB gF (X F , Z F )gF (Y F ,W F ) − gF (Y F , Z F )gF (X F ,W F ) . Let (N, h) be an m-dimensional (m ≥ 2) Riemannian manifold, and I be an open interval of the real line equipped with the negative of the standard metric. A Lorentzian manifold (M = I × f N, g = −ds2 + f 2 (s)h of dimension m+1 is a generalized Robertson-Walker space time where f : I → R+ is a smooth f (...truncated)