Narrow Results

We demonstrate that various metrizability problems for Finsler sprays can be reformulated in terms of the geodesic invariance of two tensors, namely the metric and angular tensors. We show that a spray $S$ is the geodesic spray of some Finsler metric if and only if its metric tensor is geodesically invariant. Moreover, we establish that gyroscopic sprays constitute the largest class of sprays characterized by a geodesic-invariant angular metric. Scalar functions associated with these geodesically invariant tensors will also be invariant, thereby providing first integrals for the given spray.

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A novel geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product yields two different applications. First, an intrinsic proof is provided of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. Second, some applications in geometric control theory for mechanical systems are clarified.