Cartan-Hadamard Theorem

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 $\R^n$.


This article, or a section of it, needs explaining.
In particular: What is the universal covering space of $M$? Does such a space exist? The book uses this theorem as an example of generalization of Gauss-Bonet theorem. Not all details are explained in this or any other chapter of the book, so I cannot say more.
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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
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