Affine connections for 2-osculating vector fields and geodesics

Abstract

We study the geudesics, with respect Co the connections for 2-osculating vector fields, a projections of standard horizontal vector fields. Then, we consider the wnnections on the principal bundle of affine 2-osculating frames, and on its reductions. finally, when it is possible we characterize the geodesics as the curves whose development is (an open interval of)a straight line.