On the exponential function of an ordered manifold with affine connection (Q1346318)

Let \((M,\nabla)\) be a manifold with an affine connection. If \(\Theta\) is a cone field on \(M\) that is invariant under parallel transport, we associate with \(\Theta\) a conal order \(\prec\) (causal structure) on \(M\). Under the additional assumption that \(\nabla\) has unimodular holonomy and vanishing torsion, we show that for each pair of points \(p,q \in M\) such that the order interval \([p,q] = \{r \in M : p \prec r \prec q\}\) is compact, there exists a geodesic arc from \(p\) to \(q\) whose tangents lie in the cone field. These results apply in particular to symmetric spaces and Lie groups with biinvariant connection. A major consequence is the surjectivity of the exponential function of an invariant Lie semigroup lying in a group with compact Lie algebra.