Research Article

Year 2019,
Volume: 16 Issue: 1, 81 - 97, 31.05.2019
### Abstract

### References

- R. L. Bishop, B. O’Neill, Manifolds of Negative Curvature, Trans. A.M.S., 145 (1969), 1-49.
- B. Y. Chen, Pseudo-Riemannian Geometry. δ-invariants and applications, World Scientific Publishing Co. Pte. Ltd, Usa, (2011).
- 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.
- 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.
- 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.
- K.Grime, Ch. Guenther, J. Isenberg, Second order renormalization group flow of three-dimensional homogeneous geometries, Analysis and Geometry, 21(2), (2013), 435467.
- C. Guenther, T. Oliynyk, Stability of the (two-loop) Renormalization Group flow for nonlinear sigma models. Lett. Math. Phys. 84, (2008), 149-157
- R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, (1982), 255-306.
- R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relavity, Contemporary Mathematics 71, (1986).
- 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.
- J. Lott, N. Sesum, Ricci flow on three-dimensional manifolds with symmetry, Comment. Math. Helv., 89, (2014), 132.
- W. J. Lu, Geometric flows on warped product manifold, Taiwanese J. of Math., 17 (5), (2013), 1791-1817.
- T. Marxen, Ricci, Flow on a Class of Noncompact Warped Product Manifolds, Journal of Geometric Analysis, 27, (2017), pp 134.
- T. Oliynyk, The second-order renormalization group flow for nonlinear sigma models in two dimensions, Class, Quantum Grav., 26, (2009), 8pp.
- B. O’Neill, semi-Riemannian geometry with applications to relativity, Academic Press, New York (1983).
- M. Simon, A class of riemannian manifolds that pinch when evolved by Ricci flow, Manuscr. Math., 101, (2000), 89-114.
- H. Tran, Harnak estimates for Ricci flow on a warped product, J. Geo. Anal., 24, (2015), 1-25.

Year 2019,
Volume: 16 Issue: 1, 81 - 97, 31.05.2019
### Abstract

### References

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.

- R. L. Bishop, B. O’Neill, Manifolds of Negative Curvature, Trans. A.M.S., 145 (1969), 1-49.
- B. Y. Chen, Pseudo-Riemannian Geometry. δ-invariants and applications, World Scientific Publishing Co. Pte. Ltd, Usa, (2011).
- 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.
- 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.
- 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.
- K.Grime, Ch. Guenther, J. Isenberg, Second order renormalization group flow of three-dimensional homogeneous geometries, Analysis and Geometry, 21(2), (2013), 435467.
- C. Guenther, T. Oliynyk, Stability of the (two-loop) Renormalization Group flow for nonlinear sigma models. Lett. Math. Phys. 84, (2008), 149-157
- R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, (1982), 255-306.
- R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relavity, Contemporary Mathematics 71, (1986).
- 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.
- J. Lott, N. Sesum, Ricci flow on three-dimensional manifolds with symmetry, Comment. Math. Helv., 89, (2014), 132.
- W. J. Lu, Geometric flows on warped product manifold, Taiwanese J. of Math., 17 (5), (2013), 1791-1817.
- T. Marxen, Ricci, Flow on a Class of Noncompact Warped Product Manifolds, Journal of Geometric Analysis, 27, (2017), pp 134.
- T. Oliynyk, The second-order renormalization group flow for nonlinear sigma models in two dimensions, Class, Quantum Grav., 26, (2009), 8pp.
- B. O’Neill, semi-Riemannian geometry with applications to relativity, Academic Press, New York (1983).
- M. Simon, A class of riemannian manifolds that pinch when evolved by Ricci flow, Manuscr. Math., 101, (2000), 89-114.
- H. Tran, Harnak estimates for Ricci flow on a warped product, J. Geo. Anal., 24, (2015), 1-25.

There are 17 citations in total.

Primary Language | English |
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Journal Section | Articles |

Authors | |

Publication Date | May 31, 2019 |

Published in Issue | Year 2019 Volume: 16 Issue: 1 |