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 Theorem## Cite this as:

Weisstein, Eric W. "Milnor's Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/MilnorsTheorem.html