Hypersurfaces of the spherical type degenerated





EDSGW-surfaces, surfaces of the spherical type, support function, planar lines of curvature


In this work, we dene the hypersurfaces of the spherical type degenerated (in short DST-hypersurfaces), these hypersurfaces has the geometric property that the middle spheres pass through the origin of the Euclidean space. We present a representation for these hypersurfaces in the case where the stereographic projection of the Gauss map N is given by the identity application. We characterize
the DST-hypersurfaces through a diferential equation and we give an explicit example of a two-parameter family of DST-hypersurfaces with planar lines of curvature foliated by (n-1)-dimensional spheres. Moreover, we classify the DST-hypersurfaces of rotation.


Appell P. Surfaces telles que l’origin se projette sur chaque normale au milieu des centres de curvature principaux. Amer. J. Math. 1988; 10:175–186.

Corro AV, Souza MA, Pina R. Classes Weingarten Surfaces in S2 _ R. Houston J. of Math. 2020; 46:651-664.

Dias DG, Corro AV. Classes of Generalized Weingarten Surfaces in the Euclidean 3-Space. Adv. Geom. 2016; 16(1):45–55.

Dias DG. Classes de hipersuperficies Weingarten generalizada no espaco Euclidiano[PhD thesis].[Goias]: Universidade Federal de Goias; 2014.

Ferreira W, Roitman P. Area preserving transformations in two-dimensional space forms and classical differential geometry. Israel J. Math. 2012; 190:325–348.

Jagy William C. Minimal hypersurfaces foliated by spheres. Michigan Math. J. 1991; 38(2):255–270.

Kim DS, Kim YH. Surfaces with planar lines of curvature. Honam Math. 2010; 32:777–790.

Leite ML. Surfaces with planar lines of curvature and orthogonal systems. J. Math. Anal. Appl. 2015; 421:1254–1273.

López R. Special Weingarten surfaces foliated by circles. Monatsh Math. 2008; 154:289–302.

López R. Constant mean curvature hypersurfaces foliated by spheres. Differential Geometry and its Applications. 1999; 11:245–256.

Masal’tsev LA. Surfaces with planar lines of curvature in Lobachevskii space. Izv. Vyssh. Uchebn. Zaved. Math. 2001; 3:39–46.

Musso E, Nicolodi L. Laguerre Geometry of Surfaces with Plane Lines of Curvature. Abh. Math. Sem. Univ. Hamburg. 1999; 69:123–138.

Riveros CMC, Corro AV. Hypersurfaces with planar lines of curvature in Euclidean space. Selecciones Matemáticas. 2017; 4(2):152–161.

Tenenblat K, Corro AV. Ribaucour Transformations Revisited, Communications in analysis and geometry, 2004; 12(5):1055–1082.




How to Cite

Carrion Riveros, C. M., V. Corro, A. M., & G. Dias, D. (2020). Hypersurfaces of the spherical type degenerated. Selecciones Matemáticas, 7(02), 214-221. https://doi.org/10.17268/sel.mat.2020.02.03