Note that $n$-manifolds $M$ with ${\rm Ric}\geq 0$ has a fundamental group of polynomial growth of degree $\leq n$ (proof : use Bishop volume theorem).

(Here a group $\Gamma$ is said to have polynomial growth of degree $\leq n$ if for any system of generators $S$ there is an $a> 0 $ s.t. $\phi_S(s)\leq as^n$ where $\phi_S(s)$ is the number of elements in $\Gamma$ which can be represented by words whose length is not greater than $s$. For more details, reference : Riemmnain geometry - Gallot, Hulin, and Lafontaine 148p.)

If it has $n$, then it is flat torus $T^n$. I want to know the classification wrt degree $k\leq n$ when $\pi_1(M)$ has no torsion. I think that it is a product of sphere, cylinder, torus, and so on. Is it right ?

  • 3
    $\begingroup$ No classification is known, except in low dimensions. Many more examples are known. By the way, what is a "degree"? $\endgroup$ Nov 11, 2015 at 17:16
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    $\begingroup$ @IgorBelegradek I assume the OP means the degree of the polynomial (as in "polynomial growth") $\endgroup$
    – Igor Rivin
    Nov 11, 2015 at 19:31
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    $\begingroup$ The paper arxiv.org/abs/math/0109167, together with the references on p.1, may explain why the classification is out of reach at the moment. The focus of this paper is on complete noncompact manifolds, but even for closed 1-connected manifolds there is no hope to get a classification. $\endgroup$ Nov 11, 2015 at 21:03
  • $\begingroup$ @Igor Belegradek Thank you for your comment and I will edit $\endgroup$ Nov 12, 2015 at 5:42
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    $\begingroup$ It is a major open problem if the fundamental group is finitely generated for complete connected manifolds under the Ricci nonnegative assumption. $\endgroup$
    – Misha
    Nov 12, 2015 at 23:53


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