ISOMETRIC IMMERSIONS, WITH FLAT NORMAL CONNECTION, OF DOMAINS OF $ n$-DIMENSIONAL LOBACHEVSKY SPACE INTO EUCLIDEAN SPACES. A MODEL OF A GAUGE FIELD

Abstract

One considers immersions of domains of the -dimensional space into , , that have principal directions at each point. The system of Gauss-Codazzi-Ricci equations is reduced to a certain system of equations in functions which satisfy , where the first functions are the coefficients of the line element, , of in curvature coordinates. An analytic immersion, with flat normal connection, of in is arbitrary to the extent that it depends on analytic functions of one variable.An electromagnetic field tensor is introduced in a natural way and an electric vector field and a magnetic vector field with matrix components are associated with the immersion of in . The tensor satisfies an analogue of the Maxwell equations. It is proved that the density of the topological charge is zero. This means that the inner product . The immersions with a stationary metric are considered - the analogues of monopoles. The following theorem is proved.Theorem. For any immersion of a domain of into with stationary metric, , there is one coordinate on which does not depend, and this coordinate compactifies. The immersion of a domain of can be represented as the product of a certain three-dimensional submanifold and a circle of varying radius.It is proved that there exists no regular, class , isometric immersion of the whole of into with stationary metric. Another class of immersions of into is considered for which the family of coordinate lines of curvature are geodesics. In this case is a potential field and the field does not depend on . The basic system of equations for the immersion can be reduced to a system of fewer dimensions.Certain immersions of domains of the Lobachevsky plane into are constructed that have zero Gauss torsion.Bibliography: 15 titles.