Differential geometry Riemannian and pseudo-Riemannian manifolds Concept
Riemannian manifold
Summary
A Riemannian manifold is a smooth manifold together with a Riemannian metric. It is the setting of Riemannian geometry, where one studies lengths, angles, geodesics, volume and curvature on curved spaces. Euclidean space, spheres and hyperbolic spaces are the basic examples, and every submanifold of Euclidean space inherits a Riemannian structure. Replacing positive definiteness by a Lorentzian signature gives the pseudo-Riemannian manifolds that model spacetime in general relativity.
Where this leads
52 pages build on this one.
- Berger's classification of holonomy groups
- Exponential map
- Hyperkähler manifold
- Cartan–Hadamard theorem
- Hodge star operator
- Sectional curvature
- Preissmann's theorem
- Einstein manifold
- Riemann curvature tensor
- Sphere theorem
- Bishop–Gromov inequality
- Harmonic map
- Nash embedding theorems
- Isometry of Riemannian manifolds
- Cartan's classification of symmetric spaces
- Symmetric space
- Hodge theorem
- Spin(7)-manifold
- Geometrization conjecture
- Weitzenböck identity
- Hyperbolic 3-manifold
- Soul theorem
- Bochner's formula
- Injectivity radius
- Sasakian manifold
- Instanton
- Myers's theorem
- Rauch comparison theorem
- Normal coordinates
- Chern–Gauss–Bonnet theorem
- Toponogov's theorem
- Laplace–Beltrami operator
- Riemannian distance
- Positive mass theorem
- Riemannian Penrose inequality
- Hopf–Rinow theorem
- G2 manifold
- Cheeger–Gromoll splitting theorem
- Spin structure
- Space form
- Systolic geometry
- Conjugate points
- Quaternion-Kähler manifold
- Cut locus
- Hilbert's theorem on the hyperbolic plane
- Gromov's compactness theorem
- Jacobi field
- Myers–Steenrod theorem
- Synge's theorem
- First and second variation of arc length
- Dirac operator
- Inverse mean curvature flow
Further reading
About this page
| Tag | 7CBT4SM83Y |
|---|---|
| URI | m2://7CBT4SM83Y |
| Kind | concept |
| Writing | W0 stub |
| MSC | 53C20, 53B20 |
| Specializes | Smooth manifold |
| Same as | |
| Changed | 2 November 2026 |
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What links here
Special cases
- Spin(7)-manifold
- Space form
- Einstein manifold
- Sasakian manifold
- Hermitian manifold
- Hyperbolic 3-manifold
- Quaternion-Kähler manifold
- G2 manifold
- Symmetric space
Used by
- Berger's classification of holonomy groups
- Exponential map
- Hyperkähler manifold
- Cartan–Hadamard theorem
- Hodge star operator
- Sectional curvature
- Preissmann's theorem
- Einstein manifold
- Riemann curvature tensor
- Sphere theorem
- Bishop–Gromov inequality
- Harmonic map
- Nash embedding theorems
- Isometry of Riemannian manifolds
- Cartan's classification of symmetric spaces
- Symmetric space
- Hodge theorem
- Spin(7)-manifold
- Geometrization conjecture
- Weitzenböck identity
- Hyperbolic 3-manifold
- Soul theorem
- Bochner's formula
- Injectivity radius
- Sasakian manifold
- Instanton
- Myers's theorem
- Rauch comparison theorem
- Normal coordinates
- Chern–Gauss–Bonnet theorem
- Toponogov's theorem
- Laplace–Beltrami operator
- Riemannian distance
- Positive mass theorem
- Riemannian Penrose inequality
- Hopf–Rinow theorem
- G2 manifold
- Cheeger–Gromoll splitting theorem
- Spin structure
- Space form
- Systolic geometry
- Conjugate points
- Quaternion-Kähler manifold
- Cut locus
- Hilbert's theorem on the hyperbolic plane
- Gromov's compactness theorem
- Jacobi field
- Myers–Steenrod theorem
- Synge's theorem
- First and second variation of arc length
- Dirac operator
- Inverse mean curvature flow