A lower bound for the $\Theta$ function on manifolds without conjugate points

Abstract

In this short note, we prove that the usual $\Theta$ function on a Riemannian manifold without conjugate points is uniformly bounded from below. This extends a result of Green in two dimensions. This elementary lemma implies that the Bérard remainder in the Weyl law is valid for a manifold without conjugate points, without any restriction on the dimension.