Projective Geometry on Manifolds Lecture Notes (v.0.3) | Mathematics 748b Spring 1988

According to Felix Klein's Erlanger program (1872), a (classical) geometry is the study of properties of a space X invariant under a group G of transformations of X. In practice G will be a Lie group which acts transitively on X, so that X is represented as a homogeneous space G=H, where H G is a closed subgroup. For example Euclidean geometry is the geometry of n-dimensional Euclidean space R n invariant under its group Euc(R n) of isometries (i.e. rigid motions, congruence transformations). In Euclidean geometry (or more generally any Riemannian geometry) we can speak of points, lines, parallelism of lines, angles between lines, distance between points, area, volume, etc. It is not diicult to show that all of these concepts can be derived from the notion of distance, i.e. from the metric structure of Euclidean geometry. Thus any isometry preserves all of these geometric entities. Other \weaker" geometries are obtained by removing some of these concepts. For example, similarity geometry | Euclidean geometry where the equivalence relation of congruence is replaced by the broader equivalence relation of similarity is the geometry invariant under similarity transformations | arises if one doesn't speak of distance, but does speak of angles (and lines, parallelism) etc. AAne geometry arises when one speaks only of points, lines and the relation of parallelism. And when one removes the notion of parallelism and only studies lines, points and the relation of incidence between them (e.g. three points being collinear or three lines being concurrent) one is led to projective geometry. Here is a basic example illustrating the diierences between the various geometries. Consider a particle moving along a smooth path; it has a well-deened velocity vector eld (this uses only the diierentiable structure of R n). In Euclidean geometry, it makes sense to discuss its \speed," so \motion at unit speed" (i.e. \arc-length-parametrized geodesic") is a meaningful concept there. But in aane geometry, the concept of \speed" or \arc-length" must be abandoned: yet \motion at constant speed" remains meaningful since the property of moving at constant speed can be characterized as parallelism of the velocity vector eld (zero acceleration). In projective geometry this notion of \constant speed" (or \parallel velocity") must be further broadened to the concept of \projective parameter" introduced by J.H.C. Whitehead. The development of synthetic projective geometry was begun by the French architect Desargues in 1636{1639 out of attempts to understand the …