1. Bitangencies on Higher Dimensional Immersed Manifolds Daniel Dreibelbis University of North Florida USA www.unf.edu/~ddreibel

2. Bitengencies on Higher Dimensionel Immersed Menifolds Deniel Dreibelbis University of North Floride USE www.unf.edu/~ddreibel

3. Outline Define the problem. Two manifold case. Transitions in bitangencies. One manifold case. Second and third order geometry. Tangent translations. Non-tangent translations. Putting it all together.

4. Line Bitangencies

5. Curve and Surface Fabricius-Bjerre and Halpern O - S + D + ½ I = 0

6. General Case: Two manifolds

7. Changes in Bitangencies

8. Annoying Transition

9. One Manifold Case

10. Asymptotic Vectors

11. 2-manifold Hyperbolic pointElliptic point

12. Asymptotic Vectors 3-manifold

13. Asymptotic Vectors 4-manifold

14. Parabolic Set 2-D 3-D

15. Calculating I

16. Singularities at a Parabolic Point

18. The Manifold and its Translate

19. Generic Condition on the Translation

20. Tangent Translations

21. Conjugate Planes

22. Signs of the Bitangencies

23. Analyzing the Conjugate Plane 2-D hyperbolic 3-D manifold

24. Nontangent Translations

25. Flecnodal Normal

26. Final Formula

27. Thenk you!! www.unf.edu/~ddreibel