One of the important results in Riemannian geometry is the Cheeger and Gromoll splitting theorem. It states that if a complete manifold of nonnegative Ricci curvature admits a line then it is diffeomorphic to $\Bbb R^k\times N$.
I wonder is there any splitting (tori decomposition or any) theorem for compact manifolds? any reference?
IMHO one possible statement could be something like the following:
If a compact Riemannian manifold of nonnegative Ricci curvature satisfys in "A" then it is homeomorphic to $N\# \Bbb T^m$ or $N \times \Bbb T^k$ or $N \times \Bbb S^k$.
P.S. I am interested in compact manifolds of nonnegative Ricci/sectional curvature.