LAUSR

Abstract We prove that, in any infinite dimensional or asymptotic Teichmuller space, the angles between Teichmuller geodesic rays issuing from a common point, defined by using the law of cosines, do not always exist. As a consequence, any infinite dimensional or asymptotic Teichmuller space equipped with the Teichmuller metric is not a CAT ( κ ) space for any κ ∈ R . We also establish a sufficient condition for the angles not to always exist in a Finsler manifold, and apply it to study the Hilbert metrics.