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What is the Best Riemannian Metric on a Compact Manifold?

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A Panoramic View of Riemannian Geometry
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Abstract

What is the best Riemannian structure on a given compact manifold? René Thom asked the author this question in the Strasbourg mathematics department library around 1960. I should say not only that I liked it, but also that I found it very motivating and frequently advertised it. Moreover, the question is the first problem in the problem list Yau [1296]. It is only recently that I discovered that the question of best metric was posed much earlier by Hopf in Hopf 1932 [730], page 220.

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References

  1. Surprisingly, there is an elaborate quantum field theory associated to this functional, known as topological gravity.

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  2. Minimal area would be a better name.

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  3. We know that they are finite quotients of tori by Bieberbach’s theorem 98 on page 291.

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  4. Very recently, using Hamilton’s technique with various sophisticated improvements, G. Perelman succeeded to prove Poincaré’s conjecture, see further details in section 14.4 and chapter 16, page 723.

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  5. For those unaccustomed to Turing machines, just imagine that they are computers—the notion of Turing machine is essentially the notion of computer.

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© 2003 Springer-Verlag Berlin Heidelberg

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Berger, M. (2003). What is the Best Riemannian Metric on a Compact Manifold?. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_11

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  • DOI: https://doi.org/10.1007/978-3-642-18245-7_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65317-2

  • Online ISBN: 978-3-642-18245-7

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