GEOMETRY OF SPACE-TIME AND GENERALIZED LAGRANGE GAUGE-THEORY

In sectional sign 1 and sectional sign 2 the authors present the Einstein and Maxwell equations for the generalised Lagrange space GL(n) = (M, g(ij)(x, y) = e2sigma(x,y)gamma(ij)(x)), and characterize the case of vanishing mixed curvature tensor field of the canonical linear d-connection. The Lagrangian gauge theory - in G.S. ASANOV's sense [1] is developed in sectional sign 3 for the tangent bundle endowed with (h, v)-metrics, obtaining the generalised Einstein - Yang Mills equations with respect to the metric gauge tensor fields and to the gauge field sigma(x, y) for three remarkable cases in which the metrics are derived from the fundamental tensor field g(ij)(x, y). Proofs are, in most cases, mechanical but rather tedious calculations. They are omitted.