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Journal of Lie Theory 24 (2014), No. 2, 397--419
Copyright Heldermann Verlag 2014

A Lie-Algebraic Formulation for Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces

Barbara A. Shipman
Dept. of Mathematics, University of Texas, Box 19408, Arlington, TX 76019, U.S.A.

Patrick D. Shipman
Dept. of Mathematics, Colorado State University, Box 1874, Fort Collins, CO 80523, U.S.A.

We give a Lie-algebraic formulation for the interacting geometries of orthogonal families of coordinate surfaces in 3-dimensional Euclidean- and Lorentz-orthogonal coordinate systems. A study of the Gauss-Lamé equations and their variational equations in this setting leads to formulas for constructing more general 3-dimensional coordinate transformations. To motivate the general constructions, we begin with special cases of orthogonal coordinate systems in 3-dimensional Lorentz space, built from orthogonal systems in the plane.

Keywords: Orthogonal coordinate systems, Gauss-Lame equations, Lorentz space.

MSC: 53C21, 53C12, 53A05, 53A35, 53Z05

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