# Cartan-Hadamard Theorem

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## Theorem

Let $M$ be a complete connected $n$-dimensional Riemannian manifold.

Suppose all sectional curvatures of $M$ are less than or equal to zero.

Then the universal covering space of $M$ is diffeomorphic to $\R^n$.

## Proof

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## Source of Name

This entry was named for Élie Joseph Cartan and Jacques Salomon Hadamard.

## Sources

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