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
108 pages build on this one.
- Analytic torsion
- Local index theorem
- Berger's classification of holonomy groups
- Exponential map
- Index of the de Rham complex
- Harmonic functions on Riemannian manifolds
- Tension field
- Minakshisundaram–Pleijel zeta function
- Sacks–Uhlenbeck bubbling
- Cheng–Yau gradient estimate
- Hartman's uniqueness theorem for harmonic maps
- Hyperkähler manifold
- Cartan–Hadamard theorem
- Minakshisundaram–Pleijel asymptotic expansion
- Hodge star operator
- Sectional curvature
- Mostow rigidity theorem
- Lichnerowicz formula
- Energy of a map between Riemannian manifolds
- Yamabe problem
- Energy functional on paths
- Preissmann's theorem
- Einstein manifold
- Riemann curvature tensor
- Sphere theorem
- Sunada's theorem
- Bishop–Gromov inequality
- Gradient flow on a manifold
- Harmonic map
- Nash embedding theorems
- Uhlenbeck compactness theorem
- Isometry of Riemannian manifolds
- Cartan's classification of symmetric spaces
- Symmetric space
- Hodge theorem
- Almgren–Pitts min-max theory
- Dirac operator on a spin manifold
- Signature operator
- Spin(7)-manifold
- Eells–Sampson theorem
- Geometrization conjecture
- Yang–Mills functional
- Spectral geometry
- Weitzenböck identity
- Hyperbolic 3-manifold
- Soul theorem
- Dirac bundle
- Cheeger–Müller theorem
- Path space of a manifold
- Lichnerowicz vanishing theorem
- Hopf conjecture
- Quantum ergodicity
- Bochner's formula
- Heat kernel on a Riemannian manifold
- Injectivity radius
- Lyusternik–Fet theorem
- Quantum unique ergodicity conjecture
- McKean–Singer formula
- Sasakian manifold
- Instanton
- Myers's theorem
- Rauch comparison theorem
- Normal coordinates
- Hodge Laplacian
- Chern–Gauss–Bonnet theorem
- Atiyah–Patodi–Singer index theorem
- Spectrum of the Laplace–Beltrami operator
- Hodge decomposition
- Cheeger's inequality for the first eigenvalue
- Toponogov's theorem
- Shadowing lemma
- Margulis lemma
- Laplace–Beltrami operator
- Riemannian distance
- Positive mass theorem
- Riemannian Penrose inequality
- Hopf–Rinow theorem
- G2 manifold
- Cheeger–Gromoll splitting theorem
- Cartan's classification of bounded symmetric domains
- Spin structure
- Hyperbolic geometry
- Almgren's regularity theorem
- Seeley–DeWitt coefficients
- Space form
- Systolic geometry
- Lichnerowicz–Obata eigenvalue theorem
- Yau's conjecture on infinitely many minimal surfaces
- Closed geodesic
- Period map
- Conjugate points
- Isospectral manifolds
- Quaternion-Kähler manifold
- Cut locus
- Harmonic map heat flow
- Hodge conjecture
- Gromoll–Meyer theorem on closed geodesics
- Hilbert's theorem on the hyperbolic plane
- Gromov's compactness theorem
- Jacobi field
- Morse index theorem for geodesics
- Myers–Steenrod theorem
- Synge's theorem
- First and second variation of arc length
- Dirac operator
- Ricci flow
- Maupertuis's principle
- 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 |
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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
- Analytic torsion
- Berger's classification of holonomy groups
- Exponential map
- Harmonic functions on Riemannian manifolds
- Tension field
- Minakshisundaram–Pleijel zeta function
- Sacks–Uhlenbeck bubbling
- Cheng–Yau gradient estimate
- Hartman's uniqueness theorem for harmonic maps
- Hyperkähler manifold
- Cartan–Hadamard theorem
- Minakshisundaram–Pleijel asymptotic expansion
- Hodge star operator
- Sectional curvature
- Mostow rigidity theorem
- Lichnerowicz formula
- Energy of a map between Riemannian manifolds
- Yamabe problem
- Energy functional on paths
- Preissmann's theorem
- Einstein manifold
- Riemann curvature tensor
- Sphere theorem
- Sunada's theorem
- Bishop–Gromov inequality
- Gradient flow on a manifold
- Harmonic map
- Nash embedding theorems
- Uhlenbeck compactness theorem
- Isometry of Riemannian manifolds
- Cartan's classification of symmetric spaces
- Symmetric space
- Hodge theorem
- Almgren–Pitts min-max theory
- Dirac operator on a spin manifold
- Signature operator
- Spin(7)-manifold
- Eells–Sampson theorem
- Geometrization conjecture
- Yang–Mills functional
- Spectral geometry
- Weitzenböck identity
- Hyperbolic 3-manifold
- Soul theorem
- Dirac bundle
- Cheeger–Müller theorem
- Path space of a manifold
- Lichnerowicz vanishing theorem
- Hopf conjecture
- Quantum ergodicity
- Bochner's formula
- Heat kernel on a Riemannian manifold
- Injectivity radius
- Lyusternik–Fet theorem
- Quantum unique ergodicity conjecture
- Sasakian manifold
- Instanton
- Myers's theorem
- Rauch comparison theorem
- Normal coordinates
- Hodge Laplacian
- Chern–Gauss–Bonnet theorem
- Atiyah–Patodi–Singer index theorem
- Spectrum of the Laplace–Beltrami operator
- Cheeger's inequality for the first eigenvalue
- Toponogov's theorem
- Shadowing lemma
- Margulis lemma
- Laplace–Beltrami operator
- Riemannian distance
- Positive mass theorem
- Riemannian Penrose inequality
- Hopf–Rinow theorem
- G2 manifold
- Cheeger–Gromoll splitting theorem
- Spin structure
- Hyperbolic geometry
- Almgren's regularity theorem
- Seeley–DeWitt coefficients
- Space form
- Systolic geometry
- Lichnerowicz–Obata eigenvalue theorem
- Yau's conjecture on infinitely many minimal surfaces
- Closed geodesic
- Conjugate points
- Isospectral manifolds
- Quaternion-Kähler manifold
- Cut locus
- Harmonic map heat flow
- Gromoll–Meyer theorem on closed geodesics
- Hilbert's theorem on the hyperbolic plane
- Gromov's compactness theorem
- Jacobi field
- Morse index theorem for geodesics
- Myers–Steenrod theorem
- Synge's theorem
- First and second variation of arc length
- Dirac operator
- Ricci flow
- Maupertuis's principle
- Inverse mean curvature flow