The chain rule for differentials of smooth maps

Depends on

• The differential of a smooth map

Used by

• Polar form of the metric in normal coordinates Corollary
• Polar integration may discard the cut locus Corollary
• The differential of a diffeomorphism is an isomorphism Corollary
• Left- and right-invariant vector fields Definition
• The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
• Compact isotopic submanifolds have isomorphic normal bundles and diffeomorphic complements Example
• A negative-gradient trajectory satisfies the energy identity Lemma
• A smooth local diffeomorphism lifts canonically to the orientation double cover Lemma
• Compatible tubular charts realize a prescribed normal identification Lemma
• Every integer is realized by a map to the sphere Lemma
• Framed cobordism is an equivalence relation Lemma
• Framed points in one component of the frame bundle are framed cobordant Lemma
• Graph-diagonal transversality is exactly fixed-point nondegeneracy Lemma
• Local length comparison for a conjugate-free geodesic Lemma
• Right-trivialized differential of the Lie-group exponential Lemma
• The collapse of a regular preimage is homotopic to the original map Lemma
• The derivative map is continuous Lemma
• The diagonal of a smooth manifold is a closed embedded submanifold Lemma
• The double point locus has the expected dimension 2m-n Lemma
• The local index is independent of chart, ball and trivialization Lemma
• The velocity field of an isotopy is well defined along its image Lemma
• The weak smooth topology is independent of the chosen atlas Lemma
• Adjoint is a smooth Lie-group representation Proposition
• Commuting Lie-algebra elements have multiplicative exponentials Proposition
• Cotangent pullback is contravariantly functorial Proposition
• Exponential map is natural for Lie-group homomorphisms Proposition
• Exponential scales one-parameter subgroups Proposition
• Gradient of the distance is the outward unit radial field off the base point and the cut locus Proposition
• Maurer--Cartan form is a pointwise isomorphism and left invariant Proposition
• Pullback of covariant tensors is smooth and functorial Proposition
• Right-invariant fields carry the opposite Lie bracket Proposition
• The tangent space of a regular level set is the kernel Proposition
• The tangent-bundle construction is functorial Proposition
• Transversality is invariant under diffeomorphic change of source and target Proposition
• Baker–Campbell–Hausdorff theorem Theorem
• Conjugate points are critical values of the exponential map along the geodesic Theorem
• Differential of a Lie-group homomorphism is a Lie-algebra homomorphism Theorem
• Embedded submanifolds admit local defining submersions Theorem
• Gårding estimates for the Dolbeault Laplacian on a compact Riemann surface Theorem
• Left-invariant vector fields are complete Theorem

…and 11 more results.

Dependency tree · two levels

2 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)