Editing 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]]===

*[[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 [[surface (mathematics)|surface]]s
**[[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 (mathematics)|submersion]]
**[[Immersion (mathematics)|immersion]]
**[[Embedding]]
***[[Whitney embedding theorem]]
*[[Critical value (critical point)|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 (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]]===
*[[Killing vector field]]
*[[Myers–Steenrod theorem|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==
{{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 (general relativity)|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]]
**[[Riemann curvature tensor|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 vector space|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]]
<!--*[[obstruction tensor]]
*[[Paneitz operator]]
*[[GJMS operator]]-->
*[[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]]

[[Category:Mathematics-related lists|Differential geometry]]
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