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