Normal and conormal bundles of an embedded submanifold

Depends on

• Embedded submanifolds and slice charts
• Slice-chart restrictions form a smooth atlas
• The inclusion of an embedded submanifold is a smooth embedding
• The differential of a smooth map
• The annihilator $U^\circ\leq V^*$ of $U\leq V$ and the preannihilator ${}^\circ S\leq V$ of $S\leq V^*$

Used by

• Compatible tubular neighbourhoods agree near compact sets up to ambient isotopy Corollary
• Dual surgery sphere Definition
• Framed embedded surgery sphere Definition
• Framings of a normal bundle Definition
• Normal bundle of a formal immersion Definition
• Orientation induced on a hypersurface by a coorientation Definition
• Pontryagin–Thom collapse with specified normal data Definition
• Stable normal bundle of a compact smooth manifold Definition
• Tangential and normal projections along a Riemannian submanifold Definition
• The frame bundle of a smooth manifold Definition
• The framing sign of a zero-dimensional regular preimage Definition
• The self-intersection number of a complementary-dimensional oriented submanifold Definition
• Tubular neighbourhoods of embedded submanifolds Definition
• Whitney disk, clean Whitney disk and framed Whitney disk Definition
• Compact isotopic submanifolds have isomorphic normal bundles and diffeomorphic complements Example
• The normal line of an oriented hypersurface is trivial Example
• The orthogonal normal bundle of a submanifold is defined without a metric False statement
• An embedding into Euclidean space gives a rank-(n-m) stable normal inverse Lemma
• Compatible tubular charts realize a prescribed normal identification Lemma
• Every bordism class is realized by an embedded collapse Lemma
• Formal immersion gives the tangent normal-bundle identity Lemma
• Framed points in one component of the frame bundle are framed cobordant Lemma
• Normal bundle of the zero locus of a transverse section Lemma
• Normal push-off zeros are the self-intersection points Lemma
• Positive-codimension thickening reduces closed sources to the open case Lemma
• Stable normal data supplies framings of the surgery spheres below the middle dimension Lemma
• The framing obstruction lives in the normal bundle of the surgery sphere 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
• Transversality is equivalent to surjectivity on the normal quotient Lemma
• An oriented transverse normal bundle orients an embedded submanifold Proposition
• Assuming countable choice, an ambient metric identifies the two normal bundles Proposition
• Assuming countable choice, normal and conormal bundles are smooth vector bundles Proposition
• The Euler class of an oriented even-rank normal bundle controls self-intersection Proposition
• The zero locus of a transverse section represents the Euler dual Proposition
• The Whitney trick in the codimension-two borderline case Theorem

Dependency tree · two levels

12 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)