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Exploring Duality in Immersed Manifolds: A Study on Singularities and Sphere Bundles

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This work by Daniel Dreibelbis from the University of North Florida delves into duality concepts within immersed manifolds. It outlines definitions and generalizations of duals, highlights the singularities of dual hypersurfaces, and defines dual sphere bundles while establishing their connection to singularities. The study presents explicit examples for surfaces in 4-D and 3-manifolds in 6-D, focusing on asymptotic and binormal vectors. It examines the relationships and distinctions between these vectors to better understand manifold structures and their properties.

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Exploring Duality in Immersed Manifolds: A Study on Singularities and Sphere Bundles

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  1. Duality for Immersed Manifolds Daniel Dreibelbis University of North Florida USA

  2. Umbilic Bracelet

  3. Outline • Define 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 manifolds • Explicit examples for surfaces in 4-D and 3-manifolds in 6-D

  4. Dual Hypersurfaces

  5. Dual - Curves

  6. Dual - Surfaces

  7. Duals from Sphere Bundles

  8. Duals from Sphere Bundles

  9. Duals between Sphere Bundles

  10. Generalizing Duals

  11. Examples We Can See

  12. Curves in R4

  13. Dimension = Codimension

  14. Examples: Surfaces in R4 Asymptotic Directions vs. Binormal Directions at a point

  15. Examples: Surfaces in R4 Asymptotics Binormals

  16. Singularity Curves on the Surface

  17. Examples: 3-manifolds in R6 Asymptotic Directions vs. Binormal Directions at a point Away from inflection points, asymptotic vectors and binormal vectors are projectively equivalent.

  18. Examples: 3-manifolds in R6 At inflections, the curves may or may not be projectively equivalent.

  19. Singularity Sets on the 3-manifold

  20. Thanks! • www.unf.edu/~ddreibel

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