Immersions, submersions, and constant-rank maps

Depends on

• The rank of a smooth map at a point

Used by

• A proper injective immersion is an embedding Corollary
• A submersion is transverse to every embedded submanifold Corollary
• Embedding obstructions include all immersion normal-class obstructions Corollary
• The immersion and submersion loci are open Corollary
• Refuted: a homotopy that is immersive at every earlier time is a regular homotopy Counterexample
• Refuted: the figure-eight and the round circle are regularly homotopic as oriented immersions Counterexample
• Formal immersion between smooth manifolds Definition
• Gauss frame map of an immersion into Euclidean space Definition
• Immersed submanifolds Definition
• Local defining maps for embedded submanifolds Definition
• Regular and critical points and values Definition
• Regular homotopy of immersions Definition
• Regular oriented link diagrams Definition
• Rotation number of an immersed oriented circle in the plane Definition
• Self-transverse immersions and the double point locus Definition
• Smooth embeddings Definition
• Space of immersions and space of formal immersions Definition
• Boy's surface: an immersion of the real projective plane in three-space Example
• Coordinate inclusions are immersions and coordinate projections are submersions Example
• Immersing the circle in the plane from a formal line monomorphism Example
• Plane circle immersions of rotation number k Example
• The double point dimension count for surfaces in four- and five-space Example
• The standard sphere immersion and its normal line Example
• A double point has two disjoint embedded sheet disks meeting transversely Lemma
• A self-transverse immersion has no double points when n>2m Lemma
• An immersion into Rⁿ gives a rank-(n-m) representative of the stable normal bundle Lemma
• For compact sources the immersion condition is open in the weak smooth topology Lemma
• General linear frame spaces have the Stiefel cohomology gap Lemma
• Metastable approximation of maps by embeddings Lemma
• Regular exhaustion and Dirichlet solutions on relatively compact surface domains Lemma
• Restriction of formal-immersion data has the parametric lifting property Lemma
• Smoothing continuous families of genuine immersions Lemma
• Standard and reflected two-sphere immersions have homotopic formal data in R³ Lemma
• Stiefel-Whitney classes of the tangent bundle of real projective space Lemma
• A vector bundle projection is a surjective submersion Proposition
• Pullback of a riemannian metric is riemannian exactly for immersions Proposition
• A nonempty closed n-manifold cannot immerse in R-n for n at least one Remark
• Regular homotopy allows self-intersections but never rank drop Remark
• Sphere eversion cannot be an isotopy through embeddings Remark
• Vanishing primary and characteristic obstructions do not classify embeddings Remark

…and 3 more results.

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, Ch. 4 (standard reference, not scraped)
• Will J. Merry, Differential Geometry, Definition 6.11 (standard reference, not scraped)