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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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V. E. Zakharov

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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 2-forms 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 Gauss-Lame' 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 pseudo-Euclidean 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

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