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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
Abstract:
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.
UDC:
514
MSC: Primary
53B25,
53C42; Secondary
53C50,
78A25 Received: 08.03.1987
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