A geometric application of Lagrange multipliers: extremal compatible linear connections

The L\'evi-Civita connection of a Riemannian manifold is a metric (compatible) linear connection, uniquely determined by its vanishing torsion. It is extremal in the sense that it has minimal torsion at each point. We can extend this idea to more general spaces with more general (not necessarily quadratic) indicatrix hypersurfaces in the tangent spaces. Here, the existence of compatible linear connections on the base manifold is not guaranteed anymore, which needs to be addressed along with the intrinsic characterization of the extremal one. The first step is to provide the Riemann metrizability of the compatible linear connections. This Riemannian environment establishes a one-to-one correspondence between linear connections and their torsion tensors, also giving a way of measuring the length of the latter. The second step is to solve a hybrid conditional extremum problem at each point of the base manifold, all of whose constraint equations (compatibility equations) involve functions defined on the indicatrix hypersurface. The objective function to be minimized is a quadratic squared norm function defined on the finite dimensional fiber (vector space) of the torsion tensor bundle. We express the solution by using the method of Lagrange multipliers on function spaces point by point, we present a necessary and sufficient condition of the solvability and the solution is also given in terms of intrinsic quantities affecting the uniform size of the linear isometry groups of the indicatrices. This completes the description of differential geometric spaces admitting compatible linear connections on the base manifold, called generalized Berwald spaces, in Finsler geometry.