

International conference "Geometrical Methods in Mathematical Physics"
December 13, 2011 10:00–10:45, Moscow, Lomonosov Moscow State University






Spaces of diagonal curvature and $n$orthogonal coordinate systems
V. E. Zakharov^{} ^{} University of Arizona, Department of Mathematics

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Abstract:
We define the Riemannian space $G^{n}$ as a space of diagonal curvature
(SDC) if two conditions are satisfied:
1. There is at least one coordinate system in $G^{n}$ such that the metric
tensor is diagonal.
2. In this coordinate system the Riemann curvature tensor presented by a
symmetric metric in the space of 2forms in $TG^{n}$ is diagonal; only terms
like $R_{ik,ik}$ does not vanish.
The Lame' coefficients of the metric tensor of SDC satisfy the GaussLame'
equations. Certain classes of their solutions could be found by the use of
the "dressing method", which is generalization of the Inverse Scattering
method. In this talk we describe more general classes of SDC. Their Lame'
coefficients are presented in the explicit form containing a finite number
of arbitrary functions of one variable. In the case when $G^{n}$ is a domain
in $R^{n}$ and $R_{ij,kl}\equiv 0$, the problem of classification of SDC is
equivalent to the classical problem of $n$orthogonal coordinate description
in the flat Euclidean or pseudoEuclidean space. The Lame' coefficients of
these flat SDC are separated by imposing of a certain constraint on the set
of arbitrary functions of one variable. In this case we are able to solve
the intrinsic problem of the metric tensor description as well as display
the new curvilinear coordinate system in the explicit form.
An important special class of SDC is the class of spaces embedded in $
R^{n+m} $ with a flat normal bundle. We are able to separate these spaces by
imposing of proper reductions. The Ferapontov conjecture of density of
spaces of flat bundle connection in the space of all SDC is discussed. In
addition we show that some basic exact solutions of the Einstein equation
can be presented by spaces of diagonal curvature.
Language: English

