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
69 pages build on this one.
- Rokhlin's theorem
- Handle decomposition
- Riemannian metric
- Whitney embedding theorem
- Donaldson's theorem
- Poisson manifold
- Poincaré–Hopf theorem
- Differential of a smooth map
- h-cobordism theorem
- Seiberg–Witten invariants
- Riemannian manifold
- Frobenius theorem
- Lie bracket of vector fields
- Pseudo-Riemannian manifold
- Wu's formula
- Morse homology
- Smale–Hirsch theorem
- Kervaire invariant
- Hopf degree theorem
- 11/8 conjecture
- Principal bundle
- Collar neighbourhood theorem
- Tensor field
- Symplectic manifold
- Moving frame
- Tangent bundle
- Donaldson theory
- Affine connection
- s-cobordism theorem
- Homotopy principle
- Morse lemma
- Transversality
- Hermitian manifold
- Tubular neighborhood theorem
- Submersion
- Cerf theory
- de Rham theorem
- 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
- Submanifold
- Foliation
- Whitney immersion theorem
- Pontryagin–Thom construction
- Tautological one-form
- Hirzebruch signature theorem
- Orientability
- Connection on a vector bundle
- Banach manifold
- Newlander–Nirenberg theorem
- Slice theorem
- Chow–Rashevskii theorem
- de Rham cohomology
- Cobordism ring
- Contact manifold
- G-structure
- Cobordism
- Section of a vector bundle
- Immersion
- Parallelizable manifold
- Lie derivative
Further reading
About this page
| Tag | F45VQ47GPV |
|---|---|
| URI | m2://F45VQ47GPV |
| Kind | concept |
| Writing | W0 stub |
| MSC | 58A05, 57R55 |
| Specializes | Topological manifold |
| Changed | 2 November 2026 |
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What links here
Used by
- Rokhlin's theorem
- Handle decomposition
- Riemannian metric
- Whitney embedding theorem
- Donaldson's theorem
- Poisson manifold
- Poincaré–Hopf theorem
- Differential of a smooth map
- h-cobordism theorem
- Seiberg–Witten invariants
- Riemannian manifold
- Frobenius theorem
- Lie bracket of vector fields
- Pseudo-Riemannian manifold
- Wu's formula
- Morse homology
- Smale–Hirsch theorem
- Kervaire invariant
- Hopf degree theorem
- 11/8 conjecture
- Principal bundle
- Collar neighbourhood theorem
- Tensor field
- Symplectic manifold
- Moving frame
- Tangent bundle
- Donaldson theory
- Affine connection
- Homotopy principle
- Morse lemma
- Transversality
- Tubular neighborhood theorem
- Submersion
- Cerf theory
- de Rham theorem
- 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
- Submanifold
- Foliation
- Whitney immersion theorem
- Pontryagin–Thom construction
- Tautological one-form
- Hirzebruch signature theorem
- Orientability
- Connection on a vector bundle
- Banach manifold
- Newlander–Nirenberg theorem
- Chow–Rashevskii theorem
- de Rham cohomology
- Cobordism ring
- Contact manifold
- G-structure
- Cobordism
- Section of a vector bundle
- Immersion
- Parallelizable manifold
- Lie derivative
Results about this
Instances
Special cases
- Riemannian manifold
- Parallelizable manifold
- Almost complex manifold
- Poisson manifold
- Symplectic manifold
- Contact manifold