HIGHER ORDER EINSTEIN-SCHRÖDINGER SPACES

Abstract

In 1945 A. Einstein [6] and E. Schrödinger [10] started form a generalized Riemann space, thas is, a space M associated with a nonsymmetric tensor G_ij(x) and desired to find the set of all linear connections Gⁱ_jk(x) compatible with such a metric: G_ij/k=0 (see also [1] and [2]).The geometry of this space (M.G_ij) is called the Einsten-Schrödinger's geometry [3], [4]. The purpose of this paper is to discuss a nonsymmetric tensor field G_ij(x,y⁽¹⁾,...y^(k)), where (x,y⁽¹⁾,...y^(k)) is a point of the k-osculator bundle (Osc^kM,p,M) and to obtain the results for the Einstein-Schrödinger's geometry of the higher order in a natural case. The fundamental notions and notations concerning the oscillator bundle of the higher order are given in the papers [8], [9] and in the recent Miron's book [7] and we suppose them to be known.