Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the Sinh-Laplace equation
The relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvatu...
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| Format: | Article |
| Language: | English |
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Wiley
2005-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.1393 |
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| _version_ | 1850235120583180288 |
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| author | Paul Bracken |
| author_facet | Paul Bracken |
| author_sort | Paul Bracken |
| collection | DOAJ |
| description | The relationship between solutions of the sinh-Laplace
equation and the determination of various kinds of
surfaces of constant Gaussian curvature, both positive
and negative, will be investigated here. It is shown
that when the metric is given in a particular set
of coordinates, the Gaussian curvature is related to
the sinh-Laplace equation in a direct way. The
fundamental equations of surface theory are found to
yield a type of geometrically based Lax pair for the system.
Given a particular solution of the sinh-Laplace
equation, this Lax can be integrated to determine
the three fundamental vectors related to the surface.
These are also used to determine the coordinate vector
of the surface. Some specific examples of this procedure
will be given. |
| format | Article |
| id | doaj-art-0e287e0ceefe4db6aa2bb1708536de73 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2005-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0e287e0ceefe4db6aa2bb1708536de732025-08-20T02:02:25ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252005-01-01200591393140410.1155/IJMMS.2005.1393Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the Sinh-Laplace equationPaul Bracken0Department of Mathematics, University of Texas – Pan American, Edinburg 78541-2999, TX, USAThe relationship between solutions of the sinh-Laplace equation and the determination of various kinds of surfaces of constant Gaussian curvature, both positive and negative, will be investigated here. It is shown that when the metric is given in a particular set of coordinates, the Gaussian curvature is related to the sinh-Laplace equation in a direct way. The fundamental equations of surface theory are found to yield a type of geometrically based Lax pair for the system. Given a particular solution of the sinh-Laplace equation, this Lax can be integrated to determine the three fundamental vectors related to the surface. These are also used to determine the coordinate vector of the surface. Some specific examples of this procedure will be given.http://dx.doi.org/10.1155/IJMMS.2005.1393 |
| spellingShingle | Paul Bracken Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the Sinh-Laplace equation International Journal of Mathematics and Mathematical Sciences |
| title | Determination of surfaces in three-dimensional Minkowski and
Euclidean spaces based on solutions of the Sinh-Laplace equation |
| title_full | Determination of surfaces in three-dimensional Minkowski and
Euclidean spaces based on solutions of the Sinh-Laplace equation |
| title_fullStr | Determination of surfaces in three-dimensional Minkowski and
Euclidean spaces based on solutions of the Sinh-Laplace equation |
| title_full_unstemmed | Determination of surfaces in three-dimensional Minkowski and
Euclidean spaces based on solutions of the Sinh-Laplace equation |
| title_short | Determination of surfaces in three-dimensional Minkowski and
Euclidean spaces based on solutions of the Sinh-Laplace equation |
| title_sort | determination of surfaces in three dimensional minkowski and euclidean spaces based on solutions of the sinh laplace equation |
| url | http://dx.doi.org/10.1155/IJMMS.2005.1393 |
| work_keys_str_mv | AT paulbracken determinationofsurfacesinthreedimensionalminkowskiandeuclideanspacesbasedonsolutionsofthesinhlaplaceequation |