Local and global frames of a vector bundle

Depends on

• Smooth sections, local sections, and support
• Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis

Used by

• A vector bundle is trivial if and only if it has a global frame Corollary
• A closed manifold with formally plausible rank data needs positive codimension Counterexample
• Connection one form in a local frame Definition
• Gauss frame map of an immersion into Euclidean space Definition
• Holomorphic line bundles and meromorphic sections on a Riemann surface Definition
• Nondegenerate zero of a vector field Definition
• The holomorphic line bundle associated to a divisor Definition
• Vector subbundles Definition
• An open parallelizable manifold immerses in Euclidean space of equal dimension Example
• Parallelizable tori have trivial stable normal class Example
• The normal line of an oriented hypersurface is trivial Example
• The tautological line bundle over real projective space Example
• Every vector in a fibre extends to a compactly supported smooth section Lemma
• Formal immersions of the circle in the plane are classified by the winding number Lemma
• The pullback of a trivial smooth vector bundle is canonically trivial Lemma
• A smooth distribution is exactly a locally framed constant-rank family of tangent spaces Proposition
• Local frames and local trivializations are equivalent data Proposition
• Smoothness of a bundle map is equivalent to smooth local matrices Proposition
• Smoothness of a section is equivalent to smooth local components Proposition
• A flat connection admits local parallel frames Theorem
• Chern connection of a Hermitian holomorphic line bundle Theorem
• Every smooth vector bundle admits a smooth bundle metric Theorem
• The map defined by a base-point-free linear system 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)