# Bonnet-Myers Theorem

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## Theorem

Let $M$ be a complete connected Riemannian manifold.

Suppose all the sectional curvatures of $M$ are bounded below by a positive constant.

Then $M$ is compact and has a finite fundamental group.

## Proof

## Source of Name

This entry was named for Pierre Ossian Bonnet and Sumner Byron Myers.

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 1$: What Is Curvature? Curvature in Higher Dimensions