1988 Seminar on Lie Geometry
Universidad Central de Venezuela

Lectures by Fillmore


Pages of notes are displayed at left.


        The Lie quadric
        - three figures from course subsequently given at UCSD in 1990
        - fig1  fig2  fig3

        1988 Seminar

 1- 4 : Standing assuptions
        Main example
        The hypersphere
        Homogeneous coördinates

 5- 7 : Projective setting
        Projective space
        The Lie quadric
        Oriented sphere or Lie cycle
        Polars
        Tangent prime
        Projective extensions of familiar spaces

 8-12 : Stereographic projection
        Möbius quadric
        Pole and polar
        Tangency
        Oriented contact
        Parbolic pencil
        Cones
        Projection of cones
        - figure

   13 : The Apollonius contact problem

14-17 : Relative power
        Cross ratio
        Bunches of Lie cycles

18-24 : Lie contact inversion
        Appendix - Verification of an identity
        Remark - row of cycles
        Some pictures for  n = 2
        - classical Möbius inversion
        - Möbius inversion of cycles
        - Lie contact inversion of cycles
        - second "envelope"
        Lie contact inversion - continued
        - pencil of contact inversions

25-30 : The group of Lie geometry
        Matrix representation
        The Lie quadric as homogeneous space
        The space of contact elements

31-34 : Lie frame
        Contact densty
        Special case (Chern)

 4 pp : (misc. related to Lie frame)

  1-5 : Stiefel manifold
        - the Möbius quadric
        - totally isotropic 2-planes
        - a second sphere
        - lines
        - Stiefel manifold
        - contact form

 9 pp : Invariants of Lie geometry
        Four point invariants
        General bunch
        Bunches
        Types of bunches
        Case of Euclidean space
        Search for unruled bunches