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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Surprisingly, there is an elaborate quantum field theory associated to this functional, known as topological gravity.
Minimal area would be a better name.
We know that they are finite quotients of tori by Bieberbach’s theorem 98 on page 291.
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.
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
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