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SPECIAL PROBLEMS OF SURFACE THEORY IN THE EUCLIDEAN 3-DIMENSIONAL SPACE

SPECIAL PROBLEMS OF SURFACE THEORY IN THE EUCLIDEAN 3-DIMENSIONAL SPACE DEMONSTRATIO MATHEMATICAVol. XXVNO 1-21992M. RochovskiSPECIAL PROBLEMS OF SURFACE THEORYIN THE EUCLIDEAN 3-DIMENSIONAL SPACE1. IntroductionIn the proof of the theorem of Hilbert which asserts thatthere does not exist an isometric immersion of the Lobachevskiplane L2 in the 3-dimensional Euclidean space E3 , as explainedin [1] is used the property of asymptotic lines on ahypothetic, complete (with respect to a distance functiondefined by a Riemannian metric of constant Gauss curvature3...K=-l) surface in E , that every two asymptotic lines whichbelong to different families have a point in common. Thisproperty, established by the argument that if there exists adiffeomorphism f of the Euclidean plane E2 referred to2coordinates (u ,u ) on open set UcEreferred to coordinates...2(v1,v2), then there exists a diffeomorphism F of Eon U,which transforms coordinate straightlines u2=const, u1=conston intersections of coordinate straightlinesv2=const,v1=const with U, together with the fact that asymptotic lineson the hypothetical surface define a Chebyshev net enable usto construct an increasing sequence of Chebyshev rectangleswhich exhaust the surface. The existence of the diffeomorphismf implies that the Jacobi determinant of the transformationv 1 =f 1 (u 1 ,u 2 ), v 2 =f 2 (u 1 ,u 2 ), f=(f1,f2), is different from zerofor every ( u ^ u http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

SPECIAL PROBLEMS OF SURFACE THEORY IN THE EUCLIDEAN 3-DIMENSIONAL SPACE

Rochovski, M.
Demonstratio Mathematica , Volume 25 (1-2): 24 – Jan 1, 1992
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References (1)

Publisher
de Gruyter
Copyright
© by M. Rochovski
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1992-1-202
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXVNO 1-21992M. RochovskiSPECIAL PROBLEMS OF SURFACE THEORYIN THE EUCLIDEAN 3-DIMENSIONAL SPACE1. IntroductionIn the proof of the theorem of Hilbert which asserts thatthere does not exist an isometric immersion of the Lobachevskiplane L2 in the 3-dimensional Euclidean space E3 , as explainedin [1] is used the property of asymptotic lines on ahypothetic, complete (with respect to a distance functiondefined by a Riemannian metric of constant Gauss curvature3...K=-l) surface in E , that every two asymptotic lines whichbelong to different families have a point in common. Thisproperty, established by the argument that if there exists adiffeomorphism f of the Euclidean plane E2 referred to2coordinates (u ,u ) on open set UcEreferred to coordinates...2(v1,v2), then there exists a diffeomorphism F of Eon U,which transforms coordinate straightlines u2=const, u1=conston intersections of coordinate straightlinesv2=const,v1=const with U, together with the fact that asymptotic lineson the hypothetical surface define a Chebyshev net enable usto construct an increasing sequence of Chebyshev rectangleswhich exhaust the surface. The existence of the diffeomorphismf implies that the Jacobi determinant of the transformationv 1 =f 1 (u 1 ,u 2 ), v 2 =f 2 (u 1 ,u 2 ), f=(f1,f2), is different from zerofor every ( u ^ u

Journal

Demonstratio Mathematica – de Gruyter

Published: Jan 1, 1992

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