Self-Conjugate Vectors of Immersed Manifolds in R6

Daniel DreibelbisUniversity of North Florida

USA

Shameless Self-promotion

• www.unf.edu/~ddreibel/research

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.

Conjugate Vectors

Special Case

Description of Conjugate Vectors

Curvature Veronese Surface

Possible Configurations

Elliptic Curves - Addition

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

Almost Normal Form

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

Self-Conjugate Vectors

Page 1

Page 167

Parabolic Set

Generic Structure of the Parabolic Set

Around a Triple Point

Through a Pinch Point

Generic Changes

A3 vectors and Morse Transitions

A3 vectors and Morse Transitions

Quadruple Point

Pinch Point Intersection

Thanks!

• www.unf.edu/~ddreibel/research