On natural operations with linear connections
Josef Janyška · Czechoslovak Mathematical Journal · 1985
Many authors such as Dodson, Radivoiovici [2], Kolaf [7,9], Oproiu [16], Puscas [17], Rybnikov [20] and others have dealt with prolongations of connections.By these we understand the rule transforming a given connection on a manifold into a connection on some prolongation of this manifold.For instance, a connection on JT-^ X or VY -^ X is associated with a connection on a fibre manifold Y -^ X.In the present paper we shall deal with the '^prolongation" of a linear connection on an arbitrary manifold in the following sense.Any linear connection on a manifold M may be identified with a principal connection on the first order frame bundle H^M{M,Ll)= inv Jl{R"'%M), iJ^ = Gl{m,R); throughout the paper m -dim M. We shall solve the problem how to construct a principal connection on the semiholonomic (or holonomic) second order frame bundle H^M (or H^M) which depends only on finite order derivatives of a given linear connection.All our constructions will be natural in the sense of the theory of categories.First we shall solve our problem using analytical methods.Then we shall show that there is a geometrical way of solution, which uses prolongations of a linear connection given by the geometrical constructions of Oproiu [16] and Kolar [7].As a consequence we obtain natural prolongations of any linear connection.This means that there is a constructed connection on H^M over a given connection on H^M, All connections on principal bundles will be principal.Our considerations are in the category C'.