duality for immersed manifolds

Duality for Immersed ManifoldsDaniel Dreibelbis

University of North FloridaUSA

Umbilic Bracelet

OutlineDefine duals and dual generalizations.Describe the singularities of duals of

hypersurfaces.Define dual sphere bundles, and connect

their singularities.Specific examples: asymptotic and binormal

vectors for immersed manifoldsExplicit examples for surfaces in 4-D and 3-

manifolds in 6-D

Dual Hypersurfaces

Dual - Curves

Dual - Surfaces

Duals from Sphere Bundles

Duals from Sphere Bundles

Duals between Sphere Bundles

Generalizing Duals

Examples We Can See

Curves in R4

Dimension = Codimension

Examples: Surfaces in R4

Asymptotic Directions vs. Binormal Directions at a point

Examples: Surfaces in R4

Asymptotics Binormals

Singularity Curves on the Surface

Examples: 3-manifolds in R6

Asymptotic Directions vs. Binormal Directions at a point

Away from inflection points, asymptotic vectors and binormalvectors are projectively equivalent.

Examples: 3-manifolds in R6

At inflections, the curves may or may not be projectively equivalent.

Singularity Sets on the 3-manifold

Thanks!www.unf.edu/~ddreibel