Revision #456 → #1616 · back to history
addedDefinition of the tangent bundle (disjoint union of tangent spaces)5e0576598898
addedElements of the tangent bundle as point-vector pairs5a6ffab80fa4
addedNatural projection of the tangent bundlefc9f5926d4a5
addedA section of the tangent bundle is a vector field; cotangent bundle is the dual1139e83385e1
addedA manifold is parallelizable iff its tangent bundle is trivial532f2afc7d5c
addedA manifold is framed iff its tangent bundle is stably trivial9cf5feeae2c0
addedThe n-sphere is framed for all n but parallelizable only for n = 1, 3, 7f310d927dde6
addedDerivative of a smooth map is a smooth map between tangent bundlesdb85d8c5d16a
addedThe dimension of the tangent bundle is twice the dimension of the manifold6d191b55950a
addedOver an open contractible subset the tangent bundle is diffeomorphic to a product3e20b35ec877
addedDefinition of a trivial tangent bundleba7328f81b5c
addedDefinition of parallelizable manifold (trivial tangent bundle)1aac008d2df7
addedThe tangent bundle of the unit circle is trivial (it is a Lie group)981d478c654d
addedTopology and smooth structure on the tangent bundle via coordinate charts7fffcf8ff523
addedThe tangent bundle as a rank-n vector bundle with Jacobian transition functionsace966443663
addedTangent bundle of R^n is trivialdf568ac8715f
addedTangent bundle of the unit circle is trivial (an infinite cylinder)114ca6406a02
addedOnly the tangent bundles of the line and circle are readily visualized6d13639c4914
addedTangent bundle of the unit sphere is nontrivial (hairy ball theorem)39d7a60c7a54
addedDefinition of a vector field as a smooth map into the tangent bundle7b84665a8f9e
addedA vector field is a section of the tangent bundle71a6144de432
addedThe set of vector fields is a module over the algebra of smooth functionsa5a3312838bd
addedDefinition of a local vector field (local section) and the sheaf they form080673d2821b
addedDifferential 1-forms as sections of the cotangent bundle97e67f8f556c
addedDefinition of the second-order tangent bundlef86d68c20709
addedRecursive definition of the k-th order tangent bundle9075109e27cd
addedHigher-order tangent bundles are domain and range for higher-order derivatives9a2ee9fe5902
addedJet bundles as bundles of jets on a manifold76592f2669d4
addedDefinition of the canonical vector field on the tangent bundle (diagonal map)582b0b4b3ee8
addedThe double tangent bundle is locally a product of a curved and a flat factor332f7bbb3b02
addedLocal coordinate expression of the canonical vector field42bbda071b5c
addedCanonical vector field as derivative of scalar multiplication at t = 0b461875070fc
addedAxiomatic characterization of the tangent bundle via the canonical vector fieldc137bd9285da
addedTangent lift of a curve into the double tangent bundle138902a6d5d5
addedVertical lift of a function50fe2b07c440
addedSMOKE2_HUMAN — preservation check94b07c1c5c05
addedSMOKE2_AI_MODERATED — carry-through checkc07713c8096d
addedThe tangent space at a point16c581e013a8
addedPointwise addition of vector fields48877e951d44
addedScalar multiplication of a vector field by a smooth function39acf9e3a67d