Mappings that preserve families of curves

This talk is a survey of some results in geometry from Darboux to Tits. First, any bijection of the plane which maps lines to lines is affine, i.e., a composition of a linear map and a translation. There are local versions of this result.

Next we consider maps of three-dimensional space which preserve two families of lines, one family of lines parallel to the y axis and the other family of lines lying in planes parallel to the xz plane and with gradients equal to the y coordinates. It is shown that these maps are affine, and that this implies (for SL(3,R)) a theorem of Tits that the morphisms of a spherical building come from the group.

A local version and extensions of this result to parabolic geometries are outlined.

Refreshments will be served at 3:45 p.m. (MATX 1115, Math Lounge).