Open Access
Translator Disclaimer
2005 Visualizing Ricci flow of manifolds of revolution
J. Hyam Rubinstein, Robert Sinclair
Experiment. Math. 14(3): 285-298 (2005).


We present numerical visualizations of Ricci flow of surfaces and three-dimensional manifolds of revolution. {\tt Ricci\_rot} is an educational tool that visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in {\small $\mathbb{R}^3$} is what makes direct visualization possible. The numerical lessons gained in developing this tool may be applicable to numerical simulation of Ricci flow of other surfaces. Similarly for simple three-dimensional manifolds like the 3-sphere, with a metric that is invariant under the action of {\small $SO(3)$} with 2-sphere orbits, the metric can be represented by a 2-sphere of revolution, where the distance to the axis of revolution represents the radius of a 2-sphere orbit. Hence we can also visualize the behaviour of such a metric under Ricci flow. We discuss briefly why surfaces and 3-manifolds of revolution remain embedded in {\small $\mathbb{R}^3$} and {\small $\mathbb{R}^4$}, respectively, under Ricci flow and finally indulge in some speculation about the idea of Ricci flow in the larger space of positive definite and indefinite metrics.


Download Citation

J. Hyam Rubinstein. Robert Sinclair. "Visualizing Ricci flow of manifolds of revolution." Experiment. Math. 14 (3) 285 - 298, 2005.


Published: 2005
First available in Project Euclid: 3 October 2005

zbMATH: 1081.53055
MathSciNet: MR2172707

Primary: 53-04 , 53C44

Keywords: mathematical visualization , neckpinch , Ricci flow

Rights: Copyright © 2005 A K Peters, Ltd.


Vol.14 • No. 3 • 2005
Back to Top