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. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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