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[edit]
Differential geometry of curves[edit]
- 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[edit]
- Theorema egregium
- Gauss–Bonnet theorem
- First fundamental form
- Second fundamental form
- Gauss–Codazzi–Mainardi equations
- Dupin indicatrix
- Asymptotic curve
- Curvature
- Types of surfaces
Foundations[edit]
Calculus on manifolds[edit]
See also multivariable calculus, list of multivariable calculus topics
Differential topology[edit]
- Diffeomorphism
- Orientability
- characteristic class
- differentiable map
- Critical value
- Lie derivative
- Hairy ball theorem
- Poincaré–Hopf theorem
- Stokes' theorem
- De Rham cohomology
- Sphere eversion
- Frobenius theorem (differential topology)
Fiber bundles[edit]
Fundamental structures[edit]
Riemannian geometry[edit]
Fundamental notions[edit]
Non-Euclidean geometry[edit]
Geodesic[edit]
- Prime geodesic
- Geodesic flow
- Exponential map (Lie theory)
- Exponential map (Riemannian geometry)
- Injectivity radius
- Geodesic deviation equation
[edit]
Riemannian submanifolds[edit]
- Gauss–Codazzi equations
- Darboux frame
- Hypersurface
- Induced metric
- Nash embedding theorem
- minimal surface
- Hsiang–Lawson's conjecture
Curvature of Riemannian manifolds[edit]
- Theorema Egregium
- Gauss–Bonnet theorem
- Gauss map
- Second fundamental form
- Curvature form
- Riemann curvature tensor
- Geodesic curvature
- Scalar curvature
- Sectional curvature
- Ricci curvature, Ricci flat
- Ricci decomposition
- Ricci flow
- Einstein manifold
- Holonomy
Theorems in Riemannian geometry[edit]
- 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[edit]
Laplace–Beltrami operator[edit]
Formulas and other tools[edit]
Related structures[edit]
- Intrinsic metric
- Pseudo-Riemannian manifold
- Sub-Riemannian manifold
- Finsler geometry
- General relativity
- G2 manifold
- Information geometry
Lie groups[edit]
Connections[edit]
- covariant derivative
- Levi-Civita connection
- parallel transport
- connection form
- Cartan connection
- connection (vector bundle)
- connection (principal bundle)
- Ehresmann connection
- curvature
- torsion (differential geometry)
Complex manifolds[edit]
- 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[edit]
- Symplectic topology
- Symplectic space
- Symplectic manifold
- Symplectic structure
- Symplectomorphism
- Contact structure
- Contact geometry
- Hamiltonian system
- Sasakian manifold
- Poisson manifold
Conformal geometry[edit]
- Möbius transformation
- Conformal map
- conformal connection
- tractor bundle
- Weyl curvature
- Weyl–Schouten theorem
- ambient construction
- Willmore energy
- Willmore flow
Index theory[edit]
- Atiyah–Singer index theorem
- de Rham cohomology
- Dolbeault cohomology
- elliptic complex
- Hodge theory
- pseudodifferential operator
Homogeneous spaces[edit]
Systolic geometry[edit]
- 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