Separations of Variables and Analytic Contractions on Two-Dimensional Hyperboloids
Tóm tắt
In this review we present recent results in the field of analytical contraction of Lie algebra in two-dimensional hyperbolic space. A complete geometric description for all possible orthogonal and nonorthogonal (related to the first order symmetries) systems of coordinates, which allow separation of variables of two-dimensional Laplace–Beltrami or Helmholtz equation on the two-sheeted (upper sheet)
$${{H}_{2}}$$
and the one-sheeted
$${{\tilde {H}}_{2}}$$
hyperboloids is given. The limiting transition between non subgroup (mostly parametric) and subgroup systems is conducted. The analytic contractions between various systems of coordinates in two hyperbolic spaces and Euclidean
$${{E}_{2}}$$
and Minkowski
$${{E}_{{1,1}}}$$
spaces are presented.
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