# Category:Riemannian Manifolds

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This category contains results about **Riemannian Manifolds**.

Definitions specific to this category can be found in Definitions/Riemannian Manifolds.

A **Riemannian manifold** is a smooth manifold on the real space $\R^n$ upon which a Riemannian metric has been imposed.

This article, or a section of it, needs explaining.In particular: What does "on the real space $\R^n$" mean? I think it just a bad wording. $\R^n$ is the image of coordinate functions which for each point on the manifold produce $n$ numbers known as the coordinates (c.f. Definition:Chart)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. |

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Riemannian Manifolds"

The following 31 pages are in this category, out of 31 total.

### C

- Cartan-Hadamard Theorem
- Characterization of Constant-Curvature Metrics
- Connected Riemannian Manifold with Restricted Exponential Map defined on Whole Tangent Space admits Minimizing Geodesic Segment
- Connected Riemannian Manifold with Restricted Exponential Map defined on Whole Tangent Space is Metrically Complete
- Connected Riemannian Manifolds with Local Isometry
- Connected Riemannian Manifolds with Local Isometry/Corollary
- Corollary of Gauss Lemma for Riemannian Manifolds