# 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
* 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

## 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