Projectively equivalent metrics on the torus

Let Riemannian metrics g and (g) over bar on a closed connected manifold M-n have the same geodesics, and suppose the eigenvalues of one metric with respect to the other are different at least at one point. We show that then the first Betti number b(1) (M-n) is not greater than n, and that if there exists a point where the eigenvalues of one metric with respect to the other are not all different, then the first Betti number b(1) (M-n) is less than n. In particular, if M-n is covered by the torus T-n, then the eigenvalues of one metric with respect to the other are different at every point. This allows us to classify such metrics on the torus and to separate variables in the equation on the eigenvalues of the Laplacian of g. (C) 2003 Elsevier B.V. All rights reserved.