Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The differential of a diffeomorphism is an isomorphism
Statement
If is a diffeomorphism, then is a linear isomorphism for every .
Facts & Assumptions
Given: A diffeomorphism and a point .
Differentials satisfy the chain rule for smooth compositions (The chain rule for differentials of smooth maps).
Proof
Apply [L1] to and .
The resulting identities show that is both a left and right inverse of .
Hence is a linear isomorphism.
Depends on
Used by
- Conjugation and the adjoint representation of a Lie group Definition
- The local fixed point index is independent of chart, ball and neighbourhood Lemma
- The local index is independent of chart, ball and trivialization Lemma
- Gradient of the distance is the outward unit radial field off the base point and the cut locus Proposition
- Hessian of distance in terms of radial jacobi fields Proposition
- Injectivity radius is the infimum of cut times Proposition
- Pointwise orientation sign of a local diffeomorphism Proposition
- Translations are diffeomorphisms and their differentials trivialize the tangent bundle Proposition
- Transversality is invariant under diffeomorphic change of source and target Proposition
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)