Self-Conjugate Vectors of
Immersed Manifolds in R6
Daniel Dreibelbis
University 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