Finsler metrizabilities and geodesic invariance (Q6543111)

The paper is concerned with various Finsler metrizability problems for sprays. A spray \(S\) is called gyroscopic if there exists a basic 2-form \(\omega\in \Lambda^2(M)\) and a Finsler structure \(\widetilde{F}\) such that the Euler-Lagrange equations \N\[\NS\left(\dfrac{\partial\widetilde{F}}{\partial y^i}\right)-\dfrac{\partial\widetilde{F}}{\partial x^i}=\omega_{ij}y^j\N\]\Nhold. The main results are the following three theorems.\N\NTheorem 1.1. A spray \(S\) is the geodesic spray of some Finsler structure \(F\) if and only if its metric tensor \(g_{ij}\) is covariant constant with respect to the induced dynamical covariant derivative: \(\nabla g_{ij}=0\). \N\NTheorem 1.2. Consider a spray \(S\) and a Finsler structure \(\widetilde{F}\). \N\N(i) The spray \(S\) is projectively related to the Finsler structure \(\widetilde{F}\) if and only if there exists a 1-homogeneous function \(P\in C^\infty(T_0M)\) such that the metric tensor \(\widetilde{g}_{ij}\) satisfies the Levi-Civita equations \N\[\N\nabla\widetilde{g}_{ij}=2P\widetilde{g}_{ij} + \dfrac{\partial P}{\partial y^i}\widetilde{g}_{kj}y^k + \dfrac{\partial P}{\partial y^j}\widetilde{g}_{ik}y^k.\N\] \N\N(ii) If the spray \(S\) is projectively related to the Finsler structure \(\widetilde{F}\), then we have the geodesic invariance condition \N\[\N\nabla\left(\frac{1}{\widetilde{F}}\left(\widetilde{g}_{ij}-\dfrac{\partial \widetilde{F}}{\partial y^i}\dfrac{\partial \widetilde{F}}{\partial y^j}\right)\right)=0.\] \N\NTheorem 1.3. A spray \(S\) is gyroscopic if and only if there exists a Finsler structure \(\widetilde{F}\) that satisfies the geodesic invariance condition above.