Holonomic extension of connections and symmetrization of jets

Taking into account motivations from the geometrization of mathematical physics, we discuss the symmetrization of higher order jets and extension of connections. First we introduce an extension of a general connection Γ on a fibered manifold Y → M into an r-th order holonomic connection and we classify all second-order holonomic extensions of Γ. Then we study the general problem of symmetrization of jets. We introduce symmetrizations of higher-order semiholonomic and nonholonomic prolongations and we describe all symmetrizations of an r-th order semiholonomic prolongation j ¯ r .