1988 - Course on Lie Geometry
Lectures by Fillmore, commentaries by Springer
Pages of notes are displayed at left.
Introduction - Geometries of circles
1- 4 : Euclidean geometry
Apollonious contact problem
Gergonne's constrution - two figures
5- 8 : Möbius geometry - inversive geometry
The Abelos of Pappus
Theorem of Miquel
9-12 : Laguerre geometry
Laguerre transformations
A recent theorem: Tyrrell and Powell
13-14 : Lie geometry
Lie transformation
Lie figure
15 : Subgeometries
Projective geometry - a brief summary
1- 7 : Projective space
Affine space
Vector space
8- 9 : Projective space (continued)
The Apollonius contact problem
10-18 : Projective transformations
Affine transformations
Transitivity
Cross ratio
19-27 : Quadratic forms
Quadrics
Intersections with lines
The polarity of a quadric
Cones
28-32 : Theorem of Pascal
The group of a quadric
Theorems of Cartan, Dieudonné, Witt
Projective orthogonal group
misc : self polar quadrilateral
synthetic construction of harmonic conjugates
figure of harmonic conjugtes and quadric
Lie Geometry - main topic
1- 7 : Equations of circles
Projective setting
Lie quadric
Tangency
8-13 : [missing]
14-17 : Theorem of Tyrrell and Powell
18-21 : The group of Lie geometry
(b) : - partial proof on simple transitivity
(c) : - partial proof from 1990
22-25 : Relative power
Bunches of Lie cycles
25-29 : Lie contact inversion
Classical Möbius inversion of circles
Lie contact inversion of cycles
(b) : - figure analysing rôle of Lie geometry
in Möbius inversion - from 1990
30-35 : Contact elements
Curves
Contact tranformations
Invariant curves
Subgeometries
(b) : - diagram of maps
29-31 : additional information on:
- contact inversion
- contact elements
- curves