On tori having poles
On a complete Riemannian manifold, a pole is a point which has no conjugate points along any geodesics through it. Given any \(\epsilon\), \(0<\epsilon <1\), there is a nonflat Riemannian metric of total volume 1 on the n-dimensional torus such that the volume \(\lambda\) of the set of all poles satisfies \(1-\epsilon \leq \lambda <1\). On the other hand, a torus is flat if there is a single pole with the additional property that the norm of every Jacobi field along every geodesic emanating from the pole is nondecreasing outside some ball centered at the pole. The proof of the latter statement uses Busemann functions.
scientific article; zbMATH DE number 3982003
scientific article