Relatives Geometries
DOI:
https://doi.org/10.17268/sel.mat.2022.02.03Keywords:
Relative hypersurface, Relative Dupin hypersurface, Isotropic geometry, Ribaucour transformationsAbstract
In this paper we consider M a fixed hypersurface in Euclidean space and we introduce two types of spaces relative to M, of type I and type II. We observe that when M is a hyperplane, the two geometries coincides with the isotropic geometry. By applying the theory to a Dupin hypersurface M, we define a relative Dupin hypersurface M of type I and type II , we provide necessary and sufficient conditions for a relative hypersurface M to be relative Dupin parameterized by relative lines of curvature, in both spaces. Moreover, we provides a relationship between the Dupin hypersurfaces locally associated to M by a Ribaucour transformation and the type II Dupin hypersurfaces relative M. We provide explicit examples of the Dupin hypersurface relative to a hyperplane, torus, S1 x Rn-1 and S2 x Rn-2, in both spaces.
References
Aydin ME, Ergut M. Isotropic geometry of graph surfaces associated with product production functions in economics. Tamkang J. Math. 2016; 47:433-443.
Berger M. Geometry II. Springer; 1987.
Berger M, Gostiaux B. Differential geometry: manifolds, curves and surfaces. Springer; 1988.
Bianchi L. Lezioni di geometria Differenziale. Terza Edicione. Nicola Zanichelli Editore; 1927.
Chen BY, Decu S, Verstraelen L. Notes on isotropic geometry of production models. Kragujevac J. Math. 2014; 38:23-33.
Corro AMV, Ferreira WP, Tenenblat K. On Ribaucour transformations for hypersurfaces. Mat. Contemp. 1999; 17:137-160.
Corro AMV, Ferreira WP, Tenenblat K. Ribaucour transformations for Constant mean curvature and linear Weingarten surfaces. Pacific Journal of Mathematics. 2003; 212(2):265-296.
Corro AMV, Ferreira WP, Tenenblat K. Minimal surfaces obtained by Ribaucour transformations. Geometriae Dedicata, Nettherlands. 2003; 96(1):117-150.
Corro AMV, Tenenblat K. Ribaucour transformation revisited. Comum. Geom. 2004; 12(5):1055-1082.
Da Silva LCB. The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces. J. Geom. 2019; 110:31. DOI: 10.1007/s00022-019-0488-9
Da Silva LCB. Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces. Tamkang Journal Of Mathematics. 2020; 31(1):1-23.
Pottmann H, Opitz K. Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces. Comput. Aided Geom. Des. 1994; 11:655-674.
Strubecker K. Differentialgeometrie des isotropen Raumes, I. Theorie der Raumkurven. Sitzungsber. Akad. Wiss. Wien. Math.- Naturw. Kl. IIa. 1941; 150:1-53.
Strubecker K. Differentialgeometrie des isotropen Raumes II. Die Flächen konstanter Relativkrümmung K = rt − s2. Math. Z. 1942; 47:743-777.
Strubecker K. Differentialgeometrie des isotropen Raumes. III. Flächentheorie. Math. Z. 1942; 48:369-427.
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