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Ruled surfaces in affine differential geometry

Ruled surfaces in affine differential geometry DEMONSTRATIO MATHEMATICAVol. XXIXNo 31996Wlodzimierz JelonekRULED SURFACES IN AFFINE DIFFERENTIAL GEOMETRY0. IntroductionThe aim of this paper is to give the detailed description of Blaschke structures induced on ruled surfaces (M, / ) in R3. The investigation of affine ruledsurfaces was started by W. Blaschke in the beginning of our century (see[B]).The description of affine ruled surfaces can be also found in the book [S-S]by P.A. Schirokov and A.P.Schirokov. The ruled extremal surfaces are described in [M-M], ruled surfaces with constant mean affine curvature H aredescribed in [D-M-M-S-V], In our paper we introduce for a ruled surface(M, f ) some local coordinates on a manifold M in which we describe theBlaschke structure (V, h, S) induced on (Af, / ) . We give a unified description of proper ruled spheres and other ruled surfaces. We also charecterizethose proper ruled surfaces which are quadrics. Although the results aremostly classical it seems to the author that it is important to underline thatruled surface is determined by a solution of a system of ordinary differentialequations of order 3 and its affine structure can be determined by means ofcoefficients of this system.1. PreliminariesLet M be a smooth, oriented and connected manifold, V a real vectorspace http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

Ruled surfaces in affine differential geometry

Jelonek, Wlodzimierz
Demonstratio Mathematica , Volume 29 (3): 8 – Jul 1, 1996
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References (4)

Publisher
de Gruyter
Copyright
© by Wlodzimierz Jelonek
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1996-0306
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXIXNo 31996Wlodzimierz JelonekRULED SURFACES IN AFFINE DIFFERENTIAL GEOMETRY0. IntroductionThe aim of this paper is to give the detailed description of Blaschke structures induced on ruled surfaces (M, / ) in R3. The investigation of affine ruledsurfaces was started by W. Blaschke in the beginning of our century (see[B]).The description of affine ruled surfaces can be also found in the book [S-S]by P.A. Schirokov and A.P.Schirokov. The ruled extremal surfaces are described in [M-M], ruled surfaces with constant mean affine curvature H aredescribed in [D-M-M-S-V], In our paper we introduce for a ruled surface(M, f ) some local coordinates on a manifold M in which we describe theBlaschke structure (V, h, S) induced on (Af, / ) . We give a unified description of proper ruled spheres and other ruled surfaces. We also charecterizethose proper ruled surfaces which are quadrics. Although the results aremostly classical it seems to the author that it is important to underline thatruled surface is determined by a solution of a system of ordinary differentialequations of order 3 and its affine structure can be determined by means ofcoefficients of this system.1. PreliminariesLet M be a smooth, oriented and connected manifold, V a real vectorspace

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 1996

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