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Projectively equivalent metrics on the torus. (English) Zbl 1051.37030

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.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53B10 Projective connections
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