Families of metrics geodesically equivalent to the analogs of the Poisson sphere (Q2738265)

Two (pseudo-)Riemannian and not proportional metrics \(g\) and \(\overline g\) on a manifold \(M\) are called geodesically equivalent if they have the same geodesics. Criterion: this is the case if and only if there exists an operator \(A\in\Gamma (\text{Hom} TM,\text{ Hom} TM)\), \(A\neq\text{const}.\), self-adjoint with respect to \(g\) and nondegenerate, such that the eigenspaces of \(A\) are integrable on \({\mathcal M}(A)\) and the members of the one-parameter family of functions \(I_c(\xi)= \text{det} (A+c \text{Id})g ((A+c \text{Id})^{-1}\xi, \xi)\) (parameter \(c)\) are in involution with respect to the symplectic structure \(\omega_g\). (Here \({\mathcal M}(A)\subset M\) is the open subset of all points where the number of distinct eigenvalues of \(A\) is locally constant, \(\omega_g\) is the pullback of the canonical structure \(\omega\) on \(TM\) with respect to the Legendre transformation corresponding to \(g.)\) In this case the metrics \(\overline g=g( (A+c \text{Id})^{-1} \xi,\xi)/ \text{det} (A+c \text{Id})\) are equivalent to \(g\).NEWLINENEWLINENEWLINEThe criterion is applied to the kinetic energy metrics of free motion of a rigid body with the result that the metric of the Poisson sphere admits a one-parameter family of geodescially equivalent metrics. Also the Clebsch case and the free motion in Minkowski space admit nontrivial equivalences.