Differential Geometry

Core idea

Complex geometry

• Complex manifold
• Holomorphic map
• Almost-complex structure
• Almost-complex map
• Integrable almost-complex structure
• Nijenhuis tensor of an almost-complex structure
• Newlander–Nirenberg theorem

Smooth manifolds, maps, and calculus

• Algebra of smooth functions
• Ambient isotopy
• Category of smooth manifolds
• Coordinate chart as an isomorphism of smooth manifolds
• Coordinate transformation law for a Riemannian metric
• Open submanifold
• Diffeomorphism groupoid 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
• C-infinity ring
• Local C-infinity-ringed space
• Full faithfulness of the smooth-function functor
• 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
• Biholomorphism group
• Category of complex manifolds
• 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
• Holomorphic isometric immersion
• 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
• The del-J operator
• Quaternionic plurisubharmonic function on a hypercomplex manifold
• HKT metric
• Local HKT potential theorem
• Hyperkähler isometric immersion
• Hyperkähler manifold
• Hyperkähler isometry
• Irreducible holomorphic symplectic manifold
• Obata connection
• Quaternion-Kähler manifold
• Quaternion-Kähler manifolds are Einstein
• Quaternionic manifold
• Quaternionic-Hermitian manifold
• Rotating hyperkähler isometry
• Twistor space of a quaternionic manifold
• Triholomorphic map
• Octonionic projective line

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
• Tangent space of a Grassmannian
• Tautological bundle on a Grassmannian

Symplectic manifolds and symplectomorphisms

• Almost symplectic manifold
• Canonical symplectic form on a cotangent bundle
• Category of symplectic manifolds and symplectic maps
• 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 manifold
• Symplectic isotopy
• Symplectic submanifold
• Symplectomorphism
• Symplectomorphism group
• Tautological one-form on a cotangent bundle

Hamiltonian mechanics

• Action-angle coordinates
• Classical phase space
• Completely integrable Hamiltonian system
• First integral
• Flux group
• Flux homomorphism
• Functions in involution
• Hamilton's equations
• Hamiltonian diffeomorphism
• Hamiltonian isotopy
• 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

• Poisson manifold
• Poisson map
• Symplectic leaf

Maslov theory

• Lagrangian Gauss map
• Maslov class of a Lagrangian submanifold
• Maslov class of the Lagrangian Grassmannian
• Maslov cycle
• Maslov index
• Relative Maslov index

Spectral and gauge-analytic geometry

• Geodesic flow
• Anosov flow
• Horocycle flow
• Clifford algebra
• Clifford module
• Differential operator between vector bundles
• Elliptic differential operator
• Hodge star operator
• Principal symbol of a differential operator
• Riemannian manifold
• Spinor bundle

Bianchi orbifolds

• Smooth effective orbifold
• Bianchi orbifold
• Cusps and ideal classes
• Bianchi volume formula
• Complete Bianchi collection

Weak equations, regularity, and relaxed models

• The h-principle