The differential of a smooth map

Depends on

• Derivations at a point and the tangent space

Used by

• Polar integration may discard the cut locus Corollary
• A bundle map with rank drop is not a formal immersion Counterexample
• A nontransverse pullback need not reproduce the rank of a foliation Counterexample
• A smooth map transverse to an embedded submanifold Definition
• Conjugation and the adjoint representation of a Lie group Definition
• Formal immersion between smooth manifolds Definition
• Gauss frame map of an immersion into Euclidean space Definition
• Nondegenerate fixed point Definition
• Nondegenerate zero of a vector field Definition
• Normal and conormal bundles of an embedded submanifold Definition
• Pullback of a cotangent vector Definition
• Rotation number of an immersed oriented circle in the plane Definition
• Smooth maps transverse to a regular foliation Definition
• The derivative map from immersions to formal immersions Definition
• The differential of a smooth real-valued function Definition
• The global differential or tangent map Definition
• The pullback of a covariant tensor field Definition
• The rank of a smooth map at a point Definition
• Transverse smooth maps Definition
• Compact isotopic submanifolds have isomorphic normal bundles and diffeomorphic complements Example
• The differential of a constant map is zero Example
• The differential is intrinsic, not merely a Jacobian matrix False statement
• A double point has two disjoint embedded sheet disks meeting transversely Lemma
• A smooth local diffeomorphism lifts canonically to the orientation double cover Lemma
• Graph-diagonal transversality is exactly fixed-point nondegeneracy Lemma
• No small subgroups in a Lie group Lemma
• Normal bundle of the zero locus of a transverse section Lemma
• Normal push-off zeros are the self-intersection points Lemma
• Pullback of a target germ by a smooth map is a well-defined source germ Lemma
• The diagonal of a smooth manifold is a closed embedded submanifold Lemma
• The differential sends derivations to derivations and is linear Lemma
• The double point locus has the expected dimension 2m-n Lemma
• The local intersection sign of graph against diagonal is sign det(I-Df) Lemma
• The normal bundle of the diagonal is canonically the tangent bundle Lemma
• The velocity field of an isotopy is well defined along its image Lemma
• Transverse submanifolds have product charts Lemma
• A vector bundle section with surjective vertical differential at every zero has a submanifold zero set Proposition
• F-relatedness is equivalent to the derivation intertwining law Proposition
• Level-set distributions are involutive Proposition
• Right-invariant fields carry the opposite Lie bracket Proposition

…and 7 more results.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
• Will J. Merry, Differential Geometry (standard reference, not scraped)
• Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)