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An example of non-embeddability of the Ricci flow

  • Mohammad SafdariEmail author
Article
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Abstract

For an evolution of metrics \((M,g_{t})\) there is a t-smooth family of embeddings \(e_{t}:M\rightarrow {\mathbb {R}}^{N}\) inducing \(g_{t}\), but in general there is no family of embeddings extending a given initial embedding \(e_{0}\). We give an example of this phenomenon when \(g_{t}\) is the evolution of \(g_{0}\) under the Ricci flow. We show that there are embeddings \(e_{0}\) inducing \(g_{0}\) which do not admit of t-smooth extensions to \(e_{t}\) inducing \(g_{t}\) for any \(t>0\). We also find hypersurfaces of \(\text {dim}>2\) that will not remain a hypersurface under Ricci flow for any positive time.

Keywords

Ricci flow Isometric embedding Hypersurfaces 

Notes

Acknowledgements

The author would like to thank Mehrdad Shahshahani and Burkhard Wilking for their help with this research.

References

  1. 1.
    Chow, B., Knopf, D.: The Ricci Flow: An Introduction. Volume 110 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. Volume 77 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2006)zbMATHGoogle Scholar
  3. 3.
    Lee, J.M.: Riemannian Manifolds. Volume 176 of Graduate Texts in Mathematics. Springer, New York (1997)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran

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