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An oriented transverse normal bundle orients an embedded submanifold
Statement
Assume . For an embedded submanifold, any two of the orientations of the ambient tangent bundle, tangent bundle, and transverse normal bundle determine the third.
Facts & Assumptions
Given: The axiom , an embedded submanifold , and orientations of any two among , , and the normal bundle .
Under , the normal bundle is a smooth vector bundle (Assuming countable choice, normal and conormal bundles are smooth vector bundles).
Its fibre is the quotient (Normal and conormal bundles of an embedded submanifold).
An orientation is a positive ray in the one-dimensional determinant line (Determinant-line orientations of finite-dimensional real vector spaces).
Proof
By [L1] and [L2], is an exact sequence of smooth vector bundles. Local frames of extended to frames of give the ordered smooth determinant-line isomorphism .
Under this isomorphism, [L3] turns any two selected positive rays into a unique third ray: tensor the tangent and normal rays to obtain the ambient ray, or choose the unique tangent or normal ray whose tensor product is the prescribed ambient ray. Smoothness is local in the adapted frames, so each resulting ray field is an orientation.
Depends on
Used by
- The diagonal self-intersection is the Euler number of the tangent bundle Corollary
- The self-intersection number of a complementary-dimensional oriented submanifold Definition
- Self-intersection of the zero section in an oriented plane bundle Example
- The normal line of an oriented hypersurface is trivial Example
- Cellular boundary coefficients are the signed trajectory counts Lemma
- Normal bundle of the zero locus of a transverse section Lemma
- Normal push-off zeros are the self-intersection points Lemma
- Orientation lines orient the continuation moduli spaces compatibly with gluing Lemma
- The normal bundle of the diagonal is canonically the tangent bundle Lemma
- The normal Thom class realizes the Poincare dual of a closed submanifold Lemma
- Unstable orientations induce orientations of the trajectory moduli spaces Lemma
- The self-intersection number is the Euler number of the normal 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
- Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
- Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)