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 Myers-Steenrod theorem
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 (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