Revision #118 → #1156 · back to history
addedRegular surface is a Riemannian manifold and Riemann surfaced5086f2ad2bc
addedSmooth map between Euclidean open sets064560ad9298
addedRegular surface0874577670d6
addedEquivalence of the three definitions78df981694eb
addedRegular surface has smooth manifold structureea3a37389c57
added2-sphere as a regular surface705c102a5043
addedTorus of revolution66a9cf931d16
addedHyperboloid of two sheets4403dad0ca62
addedHelicoid676100bf52e8
addedTangent space79121f73f19c
addedNormal space and unit normal vectors235129b7cf2d
addedIntrinsic definition of tangent vectors5cae9661c739
addedVector field as a derivation (Leibniz rule)413459942d6d
addedLie bracket of vector fields7e6ea062430d
addedVector fields form a Lie algebra4767876e29b0
addedFirst and second fundamental forms08d7e51962d0
addedCurvatures from the shape operator35c0f2bba5cf
addedPositivity of E, G and EG−F²53043eb229cc
addedEffect of negating the unit normalc52d742ad9b0
addedPrincipal curvatures as extreme normal-section curvaturesff8f86846323
addedChristoffel symbols5880da551e98
addedChristoffel symbols from the first fundamental form297cbd457f91
addedCodazzi equations4b6d7c3287e0
addedGauss equation3ddcef97a5a0
addedTheorema Egregium627ac1a41834
addedBonnet's fundamental theorem for embedded surfaces7c8ecc33fc0c
addedIsometry of surfacesaad01dd28a73
addedLength-preserving maps are isometries7253f5035877
addedIsometry from shared parametrization3e3074b22b27
addedCylinder and plane locally isometric00e643d73283
addedCatenoid and helicoid locally isometricbb77365b7c24
addedTangential vector field420a06328e5a
addedSmooth vector field52e77737c51d
addedCovariant derivative via Christoffel symbolse03c9376a552
addedCovariant derivative via orthogonal projectiona9ec570b865a
addedCovariant differentiation depends only on first fundamental form3d690d5c2a36
addedSurface of revolution836a9569c434
addedRegularity of a surface of revolution3d90a705e0a4
addedRotationally symmetric minimal surfacesc9837c4f3b66
addedGeodesics on surfaces of revolution (Clairaut)9d3ff7e93d5e
addedCurvatures of a quadric surface5e59095ac72a
addedRuled surface8ffb3ab09c36
addedVanishing Gaussian curvature of a ruled surfacec52d647ee779
addedVanishing Gaussian curvature characterizes developable surfaces6b924d956521
addedMinimal surface199ffc21f710
addedMeusnier: minimal iff mean curvature vanishes03dbe24a9cd9
addedDouglas–Radó solution of Plateau's problem99ee0e7df5f7
addedExplicit minimal surfaces605baa164177
addedOsserman: Gauss map of non-planar minimal surface is densee9be5a13c24d
addedSurface of constant curvaturec2425ad2addd
addedUnit sphere has curvature +1b6c1c52e430f
addedPlane and cylinder have curvature 0ed4091ea0d8a
addedPseudosphere has curvature −1fc2ed09f4916
addedTransitive symmetry groups of constant-curvature surfaces6feca41e3bbb
addedDevelopable surfaces have curvature 0053cacb3091f
addedRiemannian metric on a surfaceb29ae80895d9
addedShape operatore42f519f74c8
addedShape operator is self-adjoint17514bf4d2b0
addedEigenvalues are the principal curvaturesa563eeda27a4
addedDeterminant of shape operator is Gaussian curvaturef26391de617e
addedShape operator via unit normal fieldcac1d256a45d
addedEmbeddability of analytic metrics2f3ae97ed965
addedLength and energy of a pathe0c2a25a26b2
addedLength independent of parametrizationc0f8c91655ba
addedGeodesic (Euler equations)e86c1a6be5a2
addedEnergy-minimizing path is a geodesicbcd5b92f632d
addedGeodesic curvature6e59be481d7c
addedGeodesic curvature is intrinsic88bfa9f4d7f6
addedGeodesic iff geodesic curvature vanishesa3a33851b5f1
addedGeodesic iff acceleration is normalb9f0120db929
addedGaussian curvature in orthogonal coordinates28f728617d26
addedGauss's derivative formula for the anglea6fb1043ac85
addedExponential map25a0768e64ca
addedExponential map is a local diffeomorphism610eb7e2dcdd
addedDetermination of normal-coordinate Taylor seriesae3a5dc6cd44
addedGauss lemma714c01c7af9b
addedPolar form of the metricb61dec23e45f
addedGeodesic convexity of small balls543fccc0c14a
addedTheorema Egregium12d44d098c22
addedGauss–Jacobi equation5e098798df40
addedGaussian curvature via Laplace–Beltrami operator7150540bcfc0
addedIsothermal coordinatesb6c3ceccc7d5
addedLiouville's equation in isothermal coordinates85b8a771f61d
addedExistence of isothermal coordinatesb147ac3611e6
addedGauss's geodesic triangle formulaf632bfd8e51b
addedCurvature as limit of angle excess over area2d2a65d711fa
addedGauss–Bonnet theorem61abbcd5f5ab
addedPoincaré–Hopf index theorem52753ea80520
addedPositive curvature implies diffeomorphic to sphere0ef99a3a8257
addedHilbert: embedded closed surface has positive-curvature point2579d0ce245b
addedGarsia conformal isometric embeddingdca4b471a123
addedSimply connected constant-curvature surfaces8291a5aae6f3
addedDimension of moduli space of Riemann surfaces9eeb27dc34e9
addedPoincaré uniformization theoremdad464193b61
addedEuclidean cosine rule9ba7435b4701
addedFlat tori embed in E⁴ but not E³269e6daf0bde
addedIsometry group of the sphereded03c4d8a05
addedGeodesics on the sphere are great circles1d3c3c09a65a
addedSpherical cosine law4690f159aa76
addedArea of a spherical triangleab410219b2e8
addedRotations correspond to Möbius transformations8bf513a82144
addedSphere unique closed orientable surface of curvature +13038d6163dac
addedReal projective plane as a quotientb58d6cdab0b1
addedPoincaré disk metric5756a6b05557
addedSU(1,1) acts transitively on the diskf4784fc1df29
addedUnique geodesic between two points of the disk0b84e638d5db
addedHyperbolic distance formula13eaa270f13b
addedDisk is complete as a metric spaced4cb620e7316
addedHyperbolic cosine rule4d4decea971a
addedArea of a hyperbolic trianglea159e7ceb17c
addedDisk and half-plane conformally equivalent3d62a5859003
addedDisk unique simply connected surface of curvature −1bdb6175b07be
addedHyperbolic surface examples820528983709
addedCurvature under conformal change of metric4e8844493399
addedLiouville's equation for uniformizationc1af988e4ac3
addedSolvability for negative Euler characteristic (Trudinger)13a13643135c
addedRicci flow proof of uniformization278f732848f6
addedRiemannian (Levi-Civita) connection2b7a363e2d16
addedParallel vector field along a curvec862ebcaf63e
addedCurvature from parallel transport around loopsfe0b31cc161c
addedKobayashi: connection 1-form as pullback of the sphere'scecb431782d2
addedKlingenberg injectivity radius bound83de5a7125ca
addedBonnet diameter bound57d8fcc93e3c
addedCohn-Vossen rigidity of ovaloids04164ef61e8e
addedLiebmann: constant curvature closed surface is a sphere76819512cc8c
addedHopf: constant mean curvature genus-0 surface is a sphere55668c56e9db
addedCarathéodory conjecture on umbilic points85f2a5264a8d
addedZero Gaussian curvature surfaces are cylinders or planes3b71d032f8ac
addedHilbert's theorem on constant negative curvature9719cd6baed8
addedWillmore conjecture4cf2f1fcd562
addedSchmidt isoperimetric inequality on sphere and hyperbolic planec999f708879a
addedLoewner's torus systolic inequality678b1c8984b8
addedPu's inequality for the projective planeee21db517ccd