Isometric immersions, with flat normal connection, of domains of $n$-dimensional Lobachevsky space into Euclidean spaces. A model of a gauge field

Yu. A. Aminov

Sbornik. Mathematics, Tome 65 (1990) no. 2, pp. 279-303

Résumé

One considers immersions of domains of the $n$-dimensional space $L^n$ into $E^{n+m}$, $m\geqslant n-1$, that have $n$ principal directions at each point. The system of Gauss–Codazzi–Ricci equations is reduced to a certain system of equations in functions $H_1,\dots,H_{m+1}$ which satisfy $\sum_{i=1}^{m+1}H_i^2=1$, where the first $n$ functions are the coefficients of the line element, $ds^2=\sum_{i=1}^nH_i^2du_i^2$, of $L^n$ in curvature coordinates. An analytic immersion, with flat normal connection, of $L^n$ in $E^{n+m}$ is arbitrary to the extent that it depends on $nm$ analytic functions of one variable. An “electromagnetic field” tensor $F_{\mu\nu}$ is introduced in a natural way and an “electric” vector field $\mathbf E$ and a “magnetic” vector field $\mathbf H$ with matrix components are associated with the immersion of $L^4$ in $E^7$. The tensor $F_{\mu\nu}$ 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 $(\mathbf{EH}=0)$. 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 $L^4$ into $E^7$ with stationary metric$,$ $\mathbf E\equiv0,$ there is one coordinate on which $\mathbf H$ does not depend, and this coordinate “compactifies”. The immersion of a domain of $L^4$ can be represented as the product of a certain three-dimensional submanifold $F^3\subset E^5$ and a circle $S^1\subset E^2$ of varying radius. It is proved that there exists no regular, class $C^2$, isometric immersion of the whole of $L^n$ into $E^{2n-1}$ with stationary metric. Another class of immersions of $L^4$ into $E^7$ is considered for which the $u_4$ family of coordinate lines of curvature are geodesics. In this case $\mathbf E$ is a potential field and the field $\mathbf H$ does not depend on $u_4$. 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 $L^2$ into $E^4$ are constructed that have zero Gauss torsion. Bibliography: 15 titles.

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