Duality for Immersed Manifolds
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Transcript of 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