Dynamical and topological methods in the theory of geodesically equivalent metrics (Q1405612)

The authors develop a new method of studying geodesically equivalent metrics in the global setting. Let \(g,\overline{g}\) be two Riemannian metrics on a manifold \(M^n\), \(G:TM^n\to TM^n\) the fiberwise-linear mapping that canonically corresponds to \(\overline{g}\) with respect to \(g\) and \(S_k:TM^n\to TM^n\) the fiberwise-linear mapping given by the formula \[ S_k=\left(\frac{\det g}{\det\overline{g}}\right)^\frac{k+2}{n+1}\sum_{i=0}^k c_iG^{k-i+1}, \quad k=0,1,\dots,n-1 \] where \(c_0,c_1,\dots,c_n\) are coefficients of the characteristic polynomial \(\det(G-\lambda I)=c_0\lambda^0+\dots+c_n\lambda^n\). Then the functions \(I_k:TM^n\to\mathbb R\) are defined by \(I_k(x,\xi)=g(S_k(\xi),\xi)\) for \(\xi\in T_xM^n, x\in M^n\). The authors' approach is based on the following theorem: if the metrics \(g,\overline{g}\) are geodesically equivalent then the functions \(I_k\) pairwise commute. In particular, they are integrals in involution of the geodesic flow of the metric \(g\). Connections of this theorem with the orbital diffeomorphisms of Hamiltonian systems are explained. Below some examples of the results obtained in the paper via this approach are listed. The metrics \(g,\overline{g}\) are said to be strictly non-proportional at a point \(x\in M\) if \(\det (G-\lambda I)\big|_x\) has no multiple roots. 1) Let metrics \(g,\overline{g}\) on a connected manifold \(M^n\) be geodesically equivalent and strictly non-proportional at least at one point of \(M^n\). Then the metrics \(g,\overline{g}\) are strictly non-proportional almost everywhere. 2) Let \(g,\overline{g}\) be geodesically equivalent metrics on \(M^n\). If \(g\) admits a non-trivial Killing vector field then \(\overline{g}\) also admits a non-trivial Killing vector field. 3) Let \(g,\overline{g}\) be two metrics on \(M^n\). If functions \(I_0,I_1,\dots,I_{n-1}\) are functionally independent almost everywhere and commute, then metrics \(g,\overline{g}\) are geodesically equivalent. A number of results is stated by using operators \({\mathcal I}_k:C^2(M^n)\to C^0(M^n), k=0,\dots,n-1,\) that are defined in terms of operators \(S_k\), the operator \({\mathcal I}_{n-1}\) being exactly the Beltrami-Laplace operator.