Isometric submersions of Finsler manifolds (Q2719017)

In the Minkowski space the definition of isometric submersion is given and the relation between osculating ellipsoids is established. The dual map, the isometric embedding and Legendre transform is also examined. It is proved that if \(\rho:M\to N\) is an isometric submersion of Finsler spaces, then an immersed curve an \(N\) is a geodesic if and only if it's horizontal lift is a geodesic on \(M\). Finsler metrics on complex and quaternionic projective spaces are given all of whose geodesics are circles. Let \(\varphi\) be a Finsler metric on \(S^{2n+1} (S^{4n+3})\) such that its geodesics are great circles. If \(\varphi\) is invariant under the action of \(S^1(S^3)\), then the subduced metric on \(\mathbb{C} P^n\) \((\mathbb{H} P^n)\) is such that all its geodesics are circles. The relation between isometric submersion and symplectic reduction from the Hamiltonian point of view is found. In a symplectic vector space the isotropic, co-isotropic and Lagrangian subspaces are defined and the relations between them and Hamiltonian vector fields are given. It is proved, that the Finsler submersions are non curvature decreasing.