List of differential geometry topics explained

This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group 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
  • Nadirashvili surface

Foundations

Calculus on manifolds

See also multivariable calculus, list of multivariable calculus topics

• 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 (differential geometry)
  • pushforward (differential)
• jet (mathematics)
  • Contact (mathematics)
  • jet bundle
• Frobenius theorem (differential topology)
• 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 (differential topology)
  • Distribution (differential geometry)
  • 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 (mathematics)
• 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 (Lie theory)
• Exponential map (Riemannian geometry)
• 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 submanifolds

• 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

See main article: List of Lie group topics.

Connections

See main article: article and Connection (mathematics).

• covariant derivative
  • exterior covariant derivative
• Levi-Civita connection
• parallel transport
  • Development (differential geometry)
• 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 (vector bundle)
• connection (principal bundle)
• Ehresmann connection
• curvature
  • curvature form
  • holonomy, local holonomy
  • Chern–Weil homomorphism
  • Curvature vector
  • Curvature form
  • Curvature tensor
  • Cocurvature
• torsion (differential geometry)

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 (2-forms)
• 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 (mathematics)
• Bäcklund transform