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.
bshipman@uta.edu
Patrick D. Shipman
Dept. of Mathematics, Colorado State University, Box 1874, Fort Collins, CO 80523, U.S.A.
shipman@math.colostate.edu
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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