Geodesic
On some integrable cases in surface theory
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 15–1, Tome 234 (1996), pp. 65-124

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It is shown how to reformulate the Gauss–Codazzi system for a surface with arbitrary Gaussian curvature in the form of one second-order differential equation. A similar reformulation is performed for a surface with fixed mean curvature. In the cases of two-dimensional Bianchi surfaces of positive curvature, these equations correspond to the unitary reduction of the coupled Ernst system of the equations of general gravity. The theta-functional description of the corresponding geometric objects is given. Bibl. 22 titles. 
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     author = {D. A. Korotkin},
     title = {On some integrable cases in surface theory},
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     volume = {234},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1996_234_a7/}
}
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D. A. Korotkin. On some integrable cases in surface theory. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part 15–1, Tome 234 (1996), pp. 65-124. http://geodesic.mathdoc.fr/item/ZNSL_1996_234_a7/