List of differential geometry topics

Differential geometry of curves and surfaces

[Differential geometry of curves]

• List of curves topics
• Frenet–Serret formulas
• Curves in differential geometry
• Line element
• Curvature
• Radius of curvature
• Osculating circle
• Curve
• Fenchel's theorem

  [Differential geometry of surfaces]

• Theorema egregium
• Gauss–Bonnet theorem
• First fundamental form
• Second fundamental form
• Gauss–Codazzi–Mainardi equations
• Dupin indicatrix
• Asymptotic curve
• Curvature
• *Principal curvatures
• *Mean curvature
• *Gauss curvature
• *Elliptic point
• Types of surfaces
• *Minimal surface
• *Ruled surface
• *Conical surface
• *Developable surface

  Foundations

Calculus on manifolds

• Manifold
• *Differentiable manifold
• *Smooth manifold
• *Banach manifold
• *Fréchet manifold
• Tensor analysis
• *Tangent vector
• *Tangent space
• *Tangent bundle
• *Cotangent space
• *Cotangent bundle
• *Tensor
• *Tensor bundle
• *Vector field
• *Tensor field
• *Differential form
• *Exterior derivative
• *Lie derivative
• *pullback
• *pushforward
• jet
• *Contact
• *jet bundle
• Frobenius theorem
• Integral curve

  [Differential topology]

• Diffeomorphism
• *Large diffeomorphism
• Orientability
• characteristic class
• *Chern class
• *Pontrjagin class
• *spin structure
• differentiable map
• *submersion
• *immersion
• *Embedding
• **Whitney embedding theorem
• Critical value
• *Sard's theorem
• *Saddle point
• *Morse theory
• Lie derivative
• Hairy ball theorem
• Poincaré–Hopf theorem
• Stokes' theorem
• De Rham cohomology
• Sphere eversion
• Frobenius theorem
• *Distribution
• *integral curve
• *foliation
• *integrability conditions for differential systems

  Fiber bundles

• Fiber bundle
• Principal bundle
• *Frame bundle
• *Hopf bundle
• Associated bundle
• Vector bundle
• *Tangent bundle
• *Cotangent bundle
• *Line bundle
• Jet bundle

  Fundamental structures

• Sheaf
• Pseudogroup
• G-structure
• synthetic differential geometry

  [Riemannian geometry]

Fundamental notions

• Metric tensor
• Riemannian manifold
• *Pseudo-Riemannian manifold
• Levi-Civita connection

  Non-Euclidean geometry

• Non-Euclidean geometry
• Elliptic geometry
• *Spherical geometry
• *Sphere-world
• *Angle excess
• hyperbolic geometry
• * hyperbolic space
• * hyperboloid model
• * Poincaré disc model
• * Poincaré half-plane model
• * Poincaré metric
• * Angle of parallelism

  [Geodesic]

• Prime geodesic
• Geodesic flow
• Exponential map
• Exponential map
• Injectivity radius
• Geodesic deviation equation
• *Jacobi field

  Symmetric spaces (and related topics)

• Riemannian symmetric space
• *Margulis lemma
• Space form
• *Constant curvature
• *taut submanifold
• Uniformization theorem
• *Myers theorem
• *Gromov's compactness theorem

  [Riemannian submanifold]s

• Gauss–Codazzi equations
• Darboux frame
• Hypersurface
• Induced metric
• Nash embedding theorem
• minimal surface
• *Helicoid
• *Catenoid
• *Costa's minimal surface
• Hsiang–Lawson's conjecture

  [Curvature of Riemannian manifolds]

• Theorema Egregium
• Gauss–Bonnet theorem
• *Chern–Gauss–Bonnet theorem
• *Chern–Weil homomorphism
• Gauss map
• Second fundamental form
• Curvature form
• Riemann curvature tensor
• Geodesic curvature
• Scalar curvature
• Sectional curvature
• Ricci curvature, Ricci flat
• Ricci decomposition
• *Schouten tensor
• *Weyl curvature
• Ricci flow
• Einstein manifold
• Holonomy

  Theorems in Riemannian geometry

• Gauss–Bonnet theorem
• Hopf–Rinow theorem
• Cartan–Hadamard theorem
• Myers theorem
• Rauch comparison theorem
• Morse index theorem
• Synge theorem
• Weinstein theorem
• Toponogov theorem
• Sphere theorem
• Hodge theory
• Uniformization theorem
• Yamabe problem

  [Isometry]

• Killing vector field

  [Laplace–Beltrami operator]

• Hodge star operator
• Weitzenböck identity
• Laplacian operators in differential geometry

  Formulas and other tools

• List of coordinate charts
• List of formulas in Riemannian geometry
• Christoffel symbols

  Related structures

• Intrinsic metric
• Pseudo-Riemannian manifold
• Sub-Riemannian manifold
• Finsler geometry
• General relativity
• G2 manifold
• Information geometry
• *Fisher information metric

  Lie groups

Connections

• covariant derivative
• * exterior covariant derivative
• Levi-Civita connection
• parallel transport
• *Development
• connection form
• Cartan connection
• *affine connection
• *conformal connection
• *projective connection
• *method of moving frames
• *Cartan's equivalence method
• *Vierbein, tetrad
• *Cartan connection applications
• *Einstein–Cartan theory
• connection
• connection
• Ehresmann connection
• curvature
• *curvature form
• *holonomy, local holonomy
• *Chern–Weil homomorphism
• *Curvature vector
• *Curvature form
• *Curvature tensor
• *Cocurvature
• torsion

  [Complex manifolds]

• Riemann surface
• Complex projective space
• Kähler manifold
• Dolbeault operator
• CR manifold
• Stein manifold
• Almost complex structure
• Hermitian manifold
• Newlander–Nirenberg theorem
• Generalized complex manifold
• Calabi–Yau manifold
• Hyperkähler manifold
• K3 surface
• hypercomplex manifold
• Quaternion-Kähler manifold

  [Symplectic geometry]

• Symplectic topology
• Symplectic space
• Symplectic manifold
• Symplectic structure
• Symplectomorphism
• Contact structure
• Contact geometry
• Hamiltonian system
• Sasakian manifold
• Poisson manifold

  [Conformal geometry]

• Möbius transformation
• Conformal map
• conformal connection
• tractor bundle
• Weyl curvature
• Weyl–Schouten theorem
• ambient construction
• Willmore energy
• Willmore flow

  [Index theory]

• Atiyah–Singer index theorem
• de Rham cohomology
• Dolbeault cohomology
• elliptic complex
• Hodge theory
• pseudodifferential operator

  [Homogeneous spaces]

• Klein geometry, Erlangen programme
• symmetric space
• space form
• Maurer–Cartan form
• Examples
• *hyperbolic space
• *Gauss–Bolyai–Lobachevsky space
• *Grassmannian
• *Complex projective space
• *Real projective space
• *Euclidean space
• *Stiefel manifold
• *Upper half-plane
• *Sphere

  [Systolic geometry]

• Loewner's torus inequality
• Pu's inequality
• Gromov's inequality for complex projective space
• Wirtinger inequality
• Gromov's systolic inequality for essential manifolds
• Essential manifold
• Filling radius
• Filling area conjecture
• Bolza surface
• First Hurwitz triplet
• Hermite constant
• Systoles of surfaces
• Systolic freedom
• Systolic category

  Other

• Envelope
• Bäcklund transform