If a compact manifold 
has nonnegative Ricci
curvature tensor, then its fundamental group
has at most polynomial growth. On the other hand,
if 
has negative curvature, then its fundamental
group has exponential growth in the sense that 
grows exponentially, where 
is (essentially) the number of different "words"
of length 
which can be made in the fundamental group.
Milnor's Theorem
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References
Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.Referenced on Wolfram|Alpha
Milnor's TheoremCite this as:
Weisstein, Eric W. "Milnor's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MilnorsTheorem.html