Differential geometry of curves and surfaces

Differential geometry of curves

• List of curve topics
• Frenet-Serret formulas
• Curves in differential geometry
• Line element
• Curvature
• Radius of curvature
• Osculating circle
• Curve

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

Foundations

Calculus on manifolds

• 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
• Smale's paradox
• 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
• 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
• 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
• 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 theorem
• Uniformization theorem
• Yamabe problem

Isometry

• Killing vector field

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

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 connection
• tractor bundle
• Weyl curvature
• Weyl–Schouten theorem

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

Other

• Envelope (mathematics)
• Bäcklund transform