1. Self-Conjugate Vectors of Immersed Manifolds in R6
Daniel Dreibelbis
University of North Florida
USA

2. Shameless Self-promotion

3. Outline
Define conjugate and self-conjugate vectors, focusing on the case of 3-manifolds in Euclidean 6-space.
Look at connection between conjugate vectors and elliptic curves.
Classify generic structure of the parabolic set.
Classify generic transitions in a 1-parameter family of parabolic sets.

4. Conjugate Vectors

5. Special Case

6. Description of Conjugate Vectors

7. Curvature Veronese Surface

8. Possible Configurations

9. Elliptic Curves - Addition

10. Conjugate Map
The conjugate map is the sum of an order 2 point:

11. Almost Normal Form

12. Classification
Same curve can have different conjugate maps, one for each point of order 2.
j-invariant and conjugate map determines affine type of conjugate curve

13. Self-Conjugate Vectors

14. Page 1

15. Page 167

16. Parabolic Set

17. Generic Structure of the Parabolic Set

18. Around a Triple Point

19. Through a Pinch Point

20. Generic Changes

21. A3 vectors and Morse Transitions

22. A3 vectors and Morse Transitions

23. Quadruple Point

24. Pinch Point Intersection

25. Thanks!