### Abstract

We classify all Râ€�separable coordinate systems for the equations Î”_{4}Î¨=J^{4} _{i,â€‰j=1}g^{âˆ’1/2} âˆ‚_{â€‰j}(g^{1/2}g^{iâ€‰j}âˆ‚_{i}Î¨) =0 and J^{4} _{i,â€‰j=1}g^{iâ€‰j}âˆ‚_{i}Wâˆ‚_{â€‰j}W =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every Râ€�separable system for the Laplace equation Î”_{4}Î¨=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Î”_{4}Î¦=Î»Î¦ on one of the manifolds E_{4}, S_{1}Ã—S_{3}, S_{2}Ã—S_{2}, and S_{4}.

Original language | English (US) |
---|---|

Pages (from-to) | 42-50 |

Number of pages | 9 |

Journal | Journal of Mathematical Physics |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1981 |

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### Keywords

- CONFORMAL MAPPING
- COORDINATES
- HAMILTONâJACOBI EQUATION
- LAPLACE EQUATION
- METRICS
- RIEMANN SPACE
- SYMMETRY

### Cite this

*Journal of Mathematical Physics*,

*22*(1), 42-50. https://doi.org/10.1063/1.524753

**Nonorthogonal Râ€�separable coordinates for fourâ€�dimensional complex Riemannian spaces.** / Kalnins, E. G.; Miller, Willard.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 22, no. 1, pp. 42-50. https://doi.org/10.1063/1.524753

}

TY - JOUR

T1 - Nonorthogonal Râ€�separable coordinates for fourâ€�dimensional complex Riemannian spaces

AU - Kalnins, E. G.

AU - Miller, Willard

PY - 1981/1/1

Y1 - 1981/1/1

N2 - We classify all Râ€�separable coordinate systems for the equations Î”4Î¨=J4 i,â€‰j=1gâˆ’1/2 âˆ‚â€‰j(g1/2giâ€‰jâˆ‚iÎ¨) =0 and J4 i,â€‰j=1giâ€‰jâˆ‚iWâˆ‚â€‰jW =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every Râ€�separable system for the Laplace equation Î”4Î¨=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Î”4Î¦=Î»Î¦ on one of the manifolds E4, S1Ã—S3, S2Ã—S2, and S4.

AB - We classify all Râ€�separable coordinate systems for the equations Î”4Î¨=J4 i,â€‰j=1gâˆ’1/2 âˆ‚â€‰j(g1/2giâ€‰jâˆ‚iÎ¨) =0 and J4 i,â€‰j=1giâ€‰jâˆ‚iWâˆ‚â€‰jW =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every Râ€�separable system for the Laplace equation Î”4Î¨=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Î”4Î¦=Î»Î¦ on one of the manifolds E4, S1Ã—S3, S2Ã—S2, and S4.

KW - CONFORMAL MAPPING

KW - COORDINATES

KW - HAMILTONâJACOBI EQUATION

KW - LAPLACE EQUATION

KW - METRICS

KW - RIEMANN SPACE

KW - SYMMETRY

UR - http://www.scopus.com/inward/record.url?scp=17444396120&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17444396120&partnerID=8YFLogxK

U2 - 10.1063/1.524753

DO - 10.1063/1.524753

M3 - Article

AN - SCOPUS:17444396120

VL - 22

SP - 42

EP - 50

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 1

ER -