Section
Differential Geometry
Definitions and results in differential geometry, including manifolds, tangent spaces, and related structures.
Core idea
Complex geometry
July 2026 geometry expansion
Smooth manifolds, maps, and calculus
- Algebra of smooth functions
- Ambient isotopy
- Category of smooth manifolds
- Constant rank theorem
- Critical point of a smooth map
- Critical value of a smooth map
- Diffeomorphism group
- Disjoint union of smooth manifolds
- Inverse function theorem for manifolds
- Isotopy extension theorem
- Iterated tangent bundle
- Jet of a smooth map
- Mapping class group of a smooth manifold
- Product manifold
- Proper smooth map
- Rank of a smooth map
- Sard's theorem
- Smooth homotopy
- Smooth isotopy
- Space of smooth maps
- Tangent functor
- Whitney C-infinity topology
Submanifolds and transversality
- Clean intersection
- Diagonal transversality criterion
- Embedded submanifold
- Immersed submanifold
- Map transverse to a submanifold
- Neat embedding
- Neat submanifold
- Normal bundle
- Parametric transversality theorem
- Proper injective immersion is an embedding
- Regular level set
- Smooth fiber product
- Submanifold chart
- Thom transversality theorem
- Transverse intersection theorem
- Transverse preimage theorem
- Transverse smooth maps
- Transverse submanifolds
- Tubular neighborhood
- Tubular neighborhood theorem
- Whitney embedding theorem
Boundaries, corners, and cobordisms
- Bordism group
- Boundary and interior of a manifold with boundary
- Boundary chart
- Boundary defining function
- Boundary face of a manifold with corners
- Cobordism
- Cobordism category
- Cobordism equivalence relation
- Collar neighborhood theorem
- Corner stratification
- Depth of a point in a manifold with corners
- Double of a manifold with boundary
- Oriented cobordism
- Smooth manifold with boundary
- Smooth manifold with corners
- Smooth map between manifolds with boundary
- Smooth map between manifolds with corners
- Symmetric monoidal structure on the cobordism category
Partitions of unity, sheaves, and differential forms
- Boundary orientation
- Bump function
- Compactly supported de Rham cohomology
- Compactly supported differential form
- de Rham complex
- de Rham homotopy operator
- de Rham theorem
- Degree of a smooth map
- Fine sheaf
- Homotopy invariance of de Rham cohomology
- Integration of differential forms
- Mayer-Vietoris sequence for de Rham cohomology
- Orientability of a smooth manifold
- Orientation of a smooth manifold
- Poincaré lemma
- Sheaf of smooth functions
- Smooth exhaustion function
- Smooth Urysohn lemma
Complex manifolds and holomorphic geometry
- Biholomorphism
- Canonical bundle of a complex manifold
- Complex atlas
- Complex coordinate chart
- Complex submanifold
- Differential form of type (p,q)
- Dolbeault cohomology
- Dolbeault complex
- Dolbeault operators
- Holomorphic cotangent bundle
- Holomorphic line bundle
- Holomorphic section
- Holomorphic tangent bundle
- Holomorphic vector bundle
- Holomorphic vector-bundle morphism
- Riemann surface
- Sheaf of holomorphic functions
- Type decomposition of the complexified tangent bundle
Hermitian and Kähler geometry
- Almost-Hermitian manifold
- Complex torus
- Fubini–Study metric
- Fundamental 2-form of an almost-Hermitian manifold
- Hermitian manifold
- Kähler class
- Kähler form
- Kähler manifold
- Kähler metric
- Local Kähler potential
- Ricci form
- The d-c operator
- The ∂∂̄-lemma
Hodge theory
- Codifferential
- Einstein manifold
- Formal adjoint of a differential operator
- Frölicher spectral sequence
- Hard Lefschetz theorem
- Harmonic differential form
- Hodge decomposition of a compact Kähler manifold
- Hodge decomposition on a compact Riemannian manifold
- Hodge filtration
- Hodge Laplacian
- Hodge numbers
- Hodge theorem
- Hodge–Riemann bilinear relations
- Kähler identities
- Lefschetz operator
- Polarized Hodge structure
- Primitive cohomology
- Pure Hodge structure
- Ricci curvature
- Riemann curvature tensor
- Variation of Hodge structure
Special holonomy, quaternionic, and hyperkähler geometry
- Almost-quaternionic manifold
- Calabi–Yau manifold
- Calabi–Yau theorem
- Holomorphic symplectic form
- Holomorphic symplectic manifold
- Hyper-Hermitian manifold
- Hypercomplex manifold
- Hyperkähler manifold
- Irreducible holomorphic symplectic manifold
- Obata connection
- Quaternion-Kähler manifold
- Quaternion-Kähler manifolds are Einstein
- Quaternionic manifold
- Quaternionic-Hermitian manifold
- Twistor space of a quaternionic manifold
Grassmannians and linear symplectic geometry
- Coisotropic subspace
- Compatible almost-complex structure
- Grassmannian
- Isotropic Grassmannian
- Isotropic subspace
- Lagrangian Grassmannian
- Lagrangian submanifold
- Lagrangian subspace
- Linear coisotropic reduction
- Linear symplectomorphism
- Oriented Grassmannian
- Symplectic basis
- Symplectic basis theorem
- Symplectic linear map
- Symplectic orthogonal complement
- Symplectic subspace
- Symplectic vector space
- Symplectomorphism
- Tangent space of a Grassmannian
- Tautological bundle on a Grassmannian
Symplectic manifolds and symplectomorphisms
- Almost symplectic manifold
- Canonical symplectic form on a cotangent bundle
- Characteristic distribution of a coisotropic submanifold
- Characteristic foliation
- Coisotropic submanifold
- Conormal bundle is Lagrangian
- Cotangent lift
- Darboux theorem for symplectic manifolds
- Exact symplectic manifold
- Isotropic submanifold
- Lagrangian neighborhood theorem
- Liouville vector field
- Moser stability theorem
- Symplectic embedding
- Symplectic map
- Symplectic submanifold
- Tautological one-form on a cotangent bundle
Hamiltonian mechanics
- Action-angle coordinates
- Classical phase space
- Completely integrable Hamiltonian system
- First integral
- Functions in involution
- Hamilton's equations
- Hamiltonian diffeomorphism
- Hamiltonian flow
- Hamiltonian function
- Hamiltonian system
- Hamiltonian vector field
- Liouville–Arnold theorem
- Poisson algebra of smooth functions
- Poisson bracket on a symplectic manifold
- Symplectic vector field
- Time-dependent Hamiltonian system
Hamiltonian actions and reduction
- Atiyah–Guillemin–Sternberg convexity theorem
- Canonical moment map for a cotangent-lifted action
- Coadjoint orbit
- Coisotropic reduction
- Comoment map
- Equivariant moment map
- Hamiltonian Lie group action
- Hamiltonian torus action
- Infinitesimal generator of a Lie group action
- Kirillov–Kostant–Souriau symplectic form
- Marsden–Weinstein–Meyer reduction theorem
- Moment maps as Poisson maps
- Moment polytope
- Moment-map component
- Shifting trick
- Singular symplectic reduction
- Symplectic Lie group action
- Symplectic quotient
- Symplectic reduction at nonzero momentum
- Uniqueness and ambiguity of moment maps
Poisson geometry
Maslov theory
- Lagrangian Gauss map
- Maslov class of a Lagrangian submanifold
- Maslov class of the Lagrangian Grassmannian
- Maslov cycle
- Maslov index
- Relative Maslov index