Second Order Renormalization Group Flow on Warped Product Manifolds

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

References

1. R. L. Bishop, B. O’Neill, Manifolds of Negative Curvature, Trans. A.M.S., 145 (1969), 1-49.
2. B. Y. Chen, Pseudo-Riemannian Geometry. δ-invariants and applications, World Scientific Publishing Co. Pte. Ltd, Usa, (2011).
3. S. Das, K. Prabhu, S. Kar, Ricci flow of unwarped and warped product manifolds, International J. of Geometric Methods in Modern Physics, 5(7), (2010), 837-856.
4. K. Grime, Ch. Guenther, J. Isenberg, short-time existence for the second order renormalization group flow in general dimension, Proc. Am. Math. Soc., 143(10), (2015), 4397-4401.
5. K.Grime, Ch. Guenther, J. Isenberg, A geometric introduction to the two loop renormalization group flow,Journal of Fixed Point Theory and App., 14(1), (2013), 320.
6. K.Grime, Ch. Guenther, J. Isenberg, Second order renormalization group flow of three-dimensional homogeneous geometries, Analysis and Geometry, 21(2), (2013), 435467.
7. C. Guenther, T. Oliynyk, Stability of the (two-loop) Renormalization Group flow for nonlinear sigma models. Lett. Math. Phys. 84, (2008), 149-157
8. R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, (1982), 255-306.

9. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relavity, Contemporary Mathematics 71, (1986).
10. F. Hesamifard and M. M. Rezaii, Evolution of the RobertsonWalker metric under 2-loop renormalization group flow, International Journal of Modern Physics, 26(3), (2017), 1-14.
11. J. Lott, N. Sesum, Ricci flow on three-dimensional manifolds with symmetry, Comment. Math. Helv., 89, (2014), 132.
12. W. J. Lu, Geometric flows on warped product manifold, Taiwanese J. of Math., 17 (5), (2013), 1791-1817.
13. T. Marxen, Ricci, Flow on a Class of Noncompact Warped Product Manifolds, Journal of Geometric Analysis, 27, (2017), pp 134.
14. T. Oliynyk, The second-order renormalization group flow for nonlinear sigma models in two dimensions, Class, Quantum Grav., 26, (2009), 8pp.
15. B. O’Neill, semi-Riemannian geometry with applications to relativity, Academic Press, New York (1983).
16. M. Simon, A class of riemannian manifolds that pinch when evolved by Ricci flow, Manuscr. Math., 101, (2000), 89-114.
17. H. Tran, Harnak estimates for Ricci flow on a warped product, J. Geo. Anal., 24, (2015), 1-25.