Constructing complete projectively flat connections (Q2495312)

The purpose of this note is to tie up a couple of loose ends in the classical theory of linear connections. The author studies the problem if on a compact manifold with complete connection, any two points can be joined by a geodesic. The following result is proved: Let \(T^2\) be the two-dimensional torus. Then for any positive integer \(m,\) there is a complete torsion free projectively flat connection \(\nabla\) on \(T^2\) such that for any point \(p \in T^2\) there is a point \(q \in T^2\) with the property that any broken \(\nabla \)-geodesic between \(p\) and \(q\) has at least \(m\) breaks. Moreover if \(T^2\) is viewed as a Lie group in the usual manner, this connection is invariant under translations by elements of \(T^2\). Also it is shown that the answer to the following question is affirmative: For a connected open subset \(U\) of the Euclidean space \({\mathbb R}^n\) is the usual flat connection restricted to \(U\) projectively equivalent to the complete torsion free connection on \(U\)? This comes out as a special case of a more general result about connections on incomplete Riemannian manifolds: It is shown: Let \((M,g)\) be a not necessarily complete Riemannian manifold. Then there is a complete torsion free connection on \(M,\) that is projectively equivalent to the metric connection on \(M.\) In particular, any connected open subset \(M\) of the Euclidean space \({\mathbb R}^n\) has a complete torsion free connection such that the geodesics of \(\nabla \) are reparametrizations of straight line segments of \(M \subset {\mathbb R}^n.\)