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

Abstract

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.

Citation

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

Information

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

zbMATH: 1081.53055
MathSciNet: MR2172707

Subjects:
Primary: 53-04 , 53C44

Keywords: mathematical visualization , neckpinch , Ricci flow

Rights: Copyright © 2005 A K Peters, Ltd.

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