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Smoothness of a section is equivalent to smooth local components
Statement
Let be a smooth vector bundle, let be open, and let be a local frame on . A local section on is smooth if and only if there are smooth functions with
Facts & Assumptions
Given: A local frame on and a local section .
A local frame determines a local trivialization, and conversely (Local frames and local trivializations are equivalent data).
Smoothness is local on the source (Smoothness is local on the source).
Proof
By [L1], the chosen frame gives a bundle chart in which corresponds to the -th standard basis vector. Therefore exactly when .
In this chart, is smooth exactly when the coordinate map is smooth. Equivalently, each component is smooth, and the criterion is local on by [L2].
Depends on
Used by
- Cech cohomology of holomorphic sections of a line bundle on finite good covers Definition
- Connection one form in a local frame Definition
- Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
- Nondegenerate zero of a vector field Definition
- The maximal Dolbeault operator and its Hilbert adjoint on a compact Riemann surface Definition
- Self-intersection of the zero section in an oriented plane bundle Example
- The normal line of an oriented hypersurface is trivial Example
- Every vector in a fibre extends to a compactly supported smooth section Lemma
- Locally finite linear combinations of sections are smooth Lemma
- Pullback of the Thom class along a transverse section computes the Euler class Lemma
- A vector bundle section with surjective vertical differential at every zero has a submanifold zero set Proposition
- Local coordinate expression for a differential form Proposition
- Local sections of a distribution are freely generated by a local frame Proposition
- Smooth sections form a module over smooth functions Proposition
- Smoothness of a bundle map is equivalent to smooth local matrices Proposition
- Smoothness of a tensor field is equivalent to smooth coordinate components Proposition
- Smoothness of a vector field is equivalent to smooth coordinate components Proposition
- Chern connection of a Hermitian holomorphic line bundle Theorem
Dependency tree · two levels
15 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)