Projectively equivalent metrics on the torus
One says that two metrics \(g_1\) and \(g_2\) on a smooth manifold are {projectively equivalent} if they have the same unparametrized geodesics. One says that \(g_1\) and \(g_2\) are strictly nonproportional at a point \(P\) of \(M\) if the eigenvalues of \(g_1\) with respect to \(g_2\) are all distinct at \(P\). The authors show: Theorem 1: Suppose \(M\) is a connected closed \(n\)-dimensional manifold. Suppose there exist Riemannian metrics \(g_1\) and \(g_2\) on \(M\) which are projectively equivalent and strictly nonproportional at at least one point. Then the following hold: (1) The first Betti number \(b_1(M)\) is at most \(n\). (2) The fundamental group of \(M\) has a commutative subgroup of finite index. (3) If there exists a point where the metrics are not strictly nonproportional, then \(b_1(M)<n\). Let \(T^n\) be the \(n\) torus. Since \(b_1(T^n)=n\), this leads to the following: Corollary 1: Suppose that \(g_1\) and \(g_2\) are metrics on \(T^n\) which are projectively equivalent and strictly nonproportional at at least one point. Then they are strictly nonproportional at every point. The authors use Corollary 1 to describe and classify (in a certain sense) all projectively equivalent Riemannian metrics on the torus which are strictly nonproportional at at least one point. This enables the authors to separate variables in the equation for the eigenvalues of the Laplacian for such a metric.
scientific article; zbMATH DE number 2091325