Global analysis, analysis on manifolds Concept
Smooth manifold
Summary
A smooth manifold is a space that looks like a piece of near each point, glued together so that calculus makes sense on it. The gluing maps between overlapping coordinate charts are required to be infinitely differentiable. This lets one speak of smooth functions, tangent vectors, vector fields and differential forms on curved spaces such as spheres and tori. Smooth manifolds are the basic objects of differential geometry and differential topology.
Where this leads
162 pages build on this one.
- Morse–Smale system
- Rokhlin's theorem
- Handle decomposition
- Kuranishi structure
- Riemannian metric
- Horocycle flow
- Mean curvature flow
- Whitney embedding theorem
- Morse–Bott function
- Donaldson's theorem
- Lagrangian singularity
- Poisson manifold
- Whitney C-infinity topology
- Poincaré–Hopf theorem
- Differential of a smooth map
- h-cobordism theorem
- Seiberg–Witten invariants
- Riemannian manifold
- Energy of a map between Riemannian manifolds
- Frobenius theorem
- Takens's theorem
- Geodesic flow on a locally symmetric space
- Virtual fundamental class
- Spectral decomposition theorem
- Lie bracket of vector fields
- Banach bundle
- Pseudo-Riemannian manifold
- Gauge group
- Ehresmann's fibration theorem
- Thom–Mather isotopy lemmas
- Wu's formula
- Morse homology
- Smale–Hirsch theorem
- Kervaire invariant
- Jet bundle
- Elliptic complex
- Space of connections modulo gauge
- Frölicher space
- Hopf degree theorem
- 11/8 conjecture
- Global analysis
- Uhlenbeck compactness theorem
- Principal bundle
- Graded manifold
- CR manifold
- Collar neighbourhood theorem
- Signature operator
- Nijenhuis tensor
- Morse index
- Exterior differential system
- Atiyah–Bott theorem on Yang–Mills over Riemann surfaces
- Tensor field
- Ebin's slice theorem
- Infinitesimal stability
- Uhlenbeck's removable singularity theorem
- Symplectic manifold
- Multijet transversality theorem
- Parametrix
- Path space of a manifold
- Quantum ergodicity
- Moving frame
- Tangent bundle
- Donaldson theory
- Diffeology
- Pfaffian system
- Affine connection
- C-infinity ring
- s-cobordism theorem
- Schouten–Nijenhuis bracket
- Homotopy principle
- Wave front of a Legendrian map
- Morse lemma
- Transversality
- Lyusternik–Fet theorem
- Quantum unique ergodicity conjecture
- Hermitian manifold
- Fredholm map
- Nondegenerate critical point
- Germ of a smooth function
- Lie group action
- Tubular neighborhood theorem
- Real analytic manifold
- Caustic
- Atiyah–Patodi–Singer index theorem
- Lusternik–Schnirelmann theory of critical points
- Submersion
- Cerf theory
- Principal symbol of an operator on a manifold
- Stable map
- Zimmer's conjecture
- Lie group
- de Rham theorem
- Lie groupoid
- Sub-Riemannian manifold
- Partition of unity
- Regular value theorem
- Thom transversality theorem
- Finsler manifold
- Morse inequalities
- Orientation of a manifold
- Morse function
- Surgery theory
- Almost complex manifold
- Hyperbolic set
- Hilbert manifold
- Smooth four-dimensional Poincaré conjecture
- Parametric transversality theorem
- Atiyah–Bott fixed-point theorem
- Peetre theorem on local operators
- Cartan's magic formula
- Differential operator between vector bundles
- Singularity of a smooth map
- Axiom A
- Submanifold
- Foliation
- Free loop space
- Whitney immersion theorem
- Whitney's theorem on folds and cusps
- Zimmer program
- Mather's stability theorem
- Equivariant index theorem
- Pontryagin–Thom construction
- Tautological one-form
- Hirzebruch signature theorem
- Supermanifold
- Jet
- Orientability
- Morita equivalence of Lie groupoids
- Banach–Lie group
- Connection on a vector bundle
- Banach manifold
- Newlander–Nirenberg theorem
- Geodesic flow
- Pesin theory
- Thom–Mather stratified space
- Moduli space of instantons
- Slice theorem
- Chow–Rashevskii theorem
- Lie pseudogroup
- Elliptic operator on a compact manifold
- G-signature theorem
- Sinai–Ruelle–Bowen measure
- de Rham cohomology
- Cobordism ring
- Contact manifold
- G-structure
- Diffeomorphism group
- Cobordism
- Cartan's test for involutivity
- Anosov diffeomorphism
- Frölicher–Nijenhuis bracket
- Sub-Riemannian geodesic problem
- Gromoll–Meyer theorem on closed geodesics
- Section of a vector bundle
- Cartan–Kähler theorem
- Stability conjecture
- Conformal geometry
- Immersion
- Manifold of smooth mappings
- Parallelizable manifold
- Lie derivative
- Thom–Boardman singularities
Further reading
About this page
| Tag | F45VQ47GPV |
|---|---|
| URI | m2://F45VQ47GPV |
| Kind | concept |
| Writing | W0 stub |
| MSC | 58A05, 57R55 |
| Specializes | Topological manifold |
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What links here
Used by
- Rokhlin's theorem
- Handle decomposition
- Kuranishi structure
- Riemannian metric
- Mean curvature flow
- Whitney embedding theorem
- Morse–Bott function
- Donaldson's theorem
- Lagrangian singularity
- Poisson manifold
- Whitney C-infinity topology
- Poincaré–Hopf theorem
- Differential of a smooth map
- h-cobordism theorem
- Seiberg–Witten invariants
- Riemannian manifold
- Energy of a map between Riemannian manifolds
- Frobenius theorem
- Takens's theorem
- Virtual fundamental class
- Lie bracket of vector fields
- Pseudo-Riemannian manifold
- Gauge group
- Ehresmann's fibration theorem
- Thom–Mather isotopy lemmas
- Wu's formula
- Morse homology
- Smale–Hirsch theorem
- Kervaire invariant
- Jet bundle
- Elliptic complex
- Frölicher space
- Hopf degree theorem
- 11/8 conjecture
- Global analysis
- Principal bundle
- Graded manifold
- CR manifold
- Collar neighbourhood theorem
- Nijenhuis tensor
- Morse index
- Exterior differential system
- Tensor field
- Ebin's slice theorem
- Infinitesimal stability
- Symplectic manifold
- Multijet transversality theorem
- Parametrix
- Path space of a manifold
- Moving frame
- Tangent bundle
- Donaldson theory
- Diffeology
- Pfaffian system
- Affine connection
- C-infinity ring
- Schouten–Nijenhuis bracket
- Homotopy principle
- Wave front of a Legendrian map
- Morse lemma
- Transversality
- Nondegenerate critical point
- Germ of a smooth function
- Lie group action
- Tubular neighborhood theorem
- Real analytic manifold
- Caustic
- Lusternik–Schnirelmann theory of critical points
- Submersion
- Cerf theory
- Principal symbol of an operator on a manifold
- Stable map
- Zimmer's conjecture
- Lie group
- de Rham theorem
- Lie groupoid
- Sub-Riemannian manifold
- Partition of unity
- Regular value theorem
- Thom transversality theorem
- Finsler manifold
- Morse inequalities
- Orientation of a manifold
- Morse function
- Surgery theory
- Almost complex manifold
- Smooth four-dimensional Poincaré conjecture
- Parametric transversality theorem
- Atiyah–Bott fixed-point theorem
- Peetre theorem on local operators
- Cartan's magic formula
- Differential operator between vector bundles
- Singularity of a smooth map
- Submanifold
- Foliation
- Free loop space
- Whitney immersion theorem
- Whitney's theorem on folds and cusps
- Zimmer program
- Mather's stability theorem
- Pontryagin–Thom construction
- Tautological one-form
- Hirzebruch signature theorem
- Supermanifold
- Jet
- Orientability
- Morita equivalence of Lie groupoids
- Connection on a vector bundle
- Banach manifold
- Newlander–Nirenberg theorem
- Pesin theory
- Thom–Mather stratified space
- Chow–Rashevskii theorem
- Lie pseudogroup
- Elliptic operator on a compact manifold
- Sinai–Ruelle–Bowen measure
- de Rham cohomology
- Cobordism ring
- Contact manifold
- G-structure
- Diffeomorphism group
- Cobordism
- Cartan's test for involutivity
- Frölicher–Nijenhuis bracket
- Section of a vector bundle
- Cartan–Kähler theorem
- Conformal geometry
- Immersion
- Manifold of smooth mappings
- Parallelizable manifold
- Lie derivative
- Thom–Boardman singularities
Results about this
Instances
Special cases
- Riemannian manifold
- Parallelizable manifold
- Almost complex manifold
- Poisson manifold
- Symplectic manifold
- Contact manifold
- Complex manifold
- Real analytic manifold