List of differential geometry topics

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

 * 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

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

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

Connections

 * 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