Annals of Global Analysis and Geometry

, Volume 55, Issue 4, pp 681–685 | Cite as

An example of non-embeddability of the Ricci flow

  • Mohammad SafdariEmail author


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.


Ricci flow Isometric embedding Hypersurfaces 



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


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesSharif University of TechnologyTehranIran

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