Constructing Complete Projectively Flat Connections
Ralph Howard · Rocky Mountain Journal of Mathematics · 2005
. On any open subset U of the Euclidean space R n there is complete torsion free connection whose geodesics are reparameterizations of the intersections of the straight lines of R n with U . For any positive integer m there is a complete projectively at torsion free connection on the two dimensional torus such that for any point p there is another point q so that any broken geodesic from p to q has at least m breaks. This example is also homogeneous with respect to a transitive Lie group action. 1. Introduction. The propose of this note is to tie up a couple of loose ends in the classical theory of linear connections. First, in [6, p. 395], Spivak rises the question of if, on a compact manifold with complete connection, any two points can be joined by a geodesic. The answer is \ o" even when the connection is projectively at and homogeneous: Theorem 1. Let T 2 be the two dimensional torus. Then for any positive integer m there is a complete torsion free projectively ...