List of differential geometry topics
{{Short description|none}}
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]]=
=[[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=
=Fundamental structures=
[[Riemannian geometry]]
=Fundamental notions=
=Non-Euclidean geometry=
=[[Geodesic]]=
=[[Riemannian submanifold]]s=
=[[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=
=[[Isometry]]=
=[[Laplace–Beltrami operator]]=
=Formulas and other tools=
=Related structures=
Lie groups
{{main|List of Lie group topics}}
Connections
{{main article|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]]
[[Conformal geometry]]
[[Index theory]]
[[Homogeneous spaces]]
[[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