Linear Algebraic Groups (Part 2)
Posted by John Baez
This time we show how projective geometry ‘subsumes’ Euclidean, elliptic and hyperbolic geometry. It does so in two ways: the projective plane includes all 3 other planes, and its symmetry group contains their symmetry groups.
By the time we understand this, we’re almost ready to think about geometry as a subject that depends on a choice of group. But we’re also getting ready to think about algebraic geometry (for example, projective varieties).
- Lecture 2 (Sept. 27) - The road to projective geometry. Treating Euclidean, elliptic and hyperbolic geometry on an equal footing: in each case the symmetry group is a linear algebraic group of 3 × 3 matrices over a field , points are certain 1d subspaces of , and lines are certain 2d subspaces of . In projective geometry we take the symmetry group to be all of , take points to be all 1d subspaces of , and take lines to be all 2d subspaces of . It thus subsumes Euclidean, elliptic and hyperbolic geometry. In general we define projective -space, , to be the set of 1d subspaces of .