# Category:Riemannian Geometry

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

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

**Riemannian geometry** is the branch of differential geometry that studies Riemannian manifolds.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

### E

- Exponential Maps (2 P)

### I

- Induced Metrics (5 P)

### N

- Normal Neighborhoods (8 P)

### O

- Orthonormal Frames (2 P)

### P

- Pseudo-Riemannian Manifolds (empty)
- Pseudo-Riemannian Metrics (2 P)

### R

- Riemannian Metrics (2 P)

### S

- Scalar Product Spaces (6 P)

## Pages in category "Riemannian Geometry"

The following 56 pages are in this category, out of 56 total.

### A

### B

### C

- Characterization of Unit Tangent Bundle
- Conditions for Connected Riemannian Manifold to be Isometric to Quotient of Connected Riemannian Manifold by Covering Automorphism Group
- Conditions for Quotient Map from Riemannian Manifold to its Quotient by Discrete Lie Group to be Normal Riemannian Covering
- Conditions for Smooth Normal Covering Map to be Riemannian Covering
- Conditions for Subjective Smooth Submersion from Riemannian Manifold to its Orbit Space to be Riemannian Submersion
- Conditions for Subjective Smooth Submersion to be Riemannian Submersion
- Conformality is Equivalence Relation on Set of Riemannian Metrics
- Coordinate Representation of Divergence
- Coordinate Representation of Laplace-Beltrami Operator

### D

### E

- Eigenvalues of Compact Riemannian Manifold without Boundary are Nonnegative
- Element of Horizontal Space as Horizontal Lift of Vector Field
- Embedded Smooth Hypersurface from Regular Points of Smooth Function
- Equivalent Properties of Nondegenerate Symmetric Covariant 2-Tensor
- Existence and Uniqueness of Outward-Pointing Normal

### I

### L

### N

### P

- Parameter Independence of Riemannian Length of Admissible Curve
- Projection of Euclidean Space onto Euclidean Subspace is Riemannian Submersion
- Projection of Product Manifold onto Factor Manifold is Riemannian Submersion
- Projection of Warped Product Manifold onto Unwarped Factor Manifold is Riemannian Submersion
- Pullback of Riemannian Metric by Smooth Mapping is Riemannian Metric iff Mapping is Immersion

### R

- Regular Curve in Riemannian Manifold has Unit-Speed Forward Reparametrization
- Relation between Volume Forms of Conformally Related Metrics on Oriented Riemannian Manifold
- Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature
- Riemannian Manifold as Metric Space
- Riemannian Manifold has Zero Gaussian Curvature iff Euclidean
- Riemannian Volume Form of Orientable Hypersurface
- Riemannian Volume Form under Orientation-Preserving Isometry