Note that n-manifolds M with Ric≥0 has a fundamental group of polynomial growth of degree ≤n (proof : use Bishop volume theorem).

(Here a group Γ is said to have polynomial growth of degree ≤n if for any system of generators S there is an a>0 s.t. ϕS(s)≤asn where ϕS(s) is the number of elements in Γ 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 Tn. I want to know the classification wrt degree k≤n when π1(M) has no torsion. I think that it is a product of sphere, cylinder, torus, and so on. Is it right ?

**Answer**

**Attribution***Source : Link , Question Author : Hee Kwon Lee , Answer Author : Community*