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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.

C.F.G
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  • If you are asking if $M$ is isometric to a product, then it's no and an example is given here – Arctic Char Oct 20 '21 at 13:54
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    A possibly easy way to do this is to figure out what assumptions imply that a geodesic on the compact manifold lifts to a line in the universal cover and apply Cheeger-Gromoll. – Deane Oct 20 '21 at 17:13
  • @ArcticChar: I am asking if M is homeomorphic to product. is this same as that? – C.F.G Nov 07 '21 at 21:11

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