Classification of manifolds with Ric≥0{\rm Ric}\geq 0 wrt fundamental group

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 ?

Answer

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