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 and surfaces
[Differential geometry of curves]
- List of curves topics
- Frenet–Serret formulas
- Curves in differential geometry
- Line element
- Curvature
- Radius of [curvature (mathematics)|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
[Riemannian geometry]
Fundamental notions
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 (geometry)|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]
[Laplace–Beltrami operator]
Formulas and other tools
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