Journal ArticleDOI
The geometry of surfaces in 4-space from a contact viewpoint
TLDR
In this article, the authors studied the geometry of convex surfaces embedded in ℝ4 through their generic contacts with hyperplanes and proved that the inflection points on them are the umbilic points of their families of height functions.Abstract:
We study the geometry of the surfaces embedded in ℝ4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any generic convexly embedded 2-sphere in ℝ4 has inflection points.read more
Citations
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Journal ArticleDOI
Inflection points and topology of surfaces in 4-space
Ronaldo Garcia,Dirce Kiyomi Hayashida Mochida,Maria del Carmen Romero Fuster,Maria Aparecida Soares Ruas +3 more
TL;DR: In this paper, it was shown that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.
The lightcone gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space
TL;DR: The notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space was introduced in this article.
Journal ArticleDOI
Singularities of evolutes of hypersurfaces in hyperbolic space
TL;DR: In this paper, the differential geometry of hypersurfaces in hyperbolic space is studied as an application of the theory of Lagrangian singularities, and the contact of hypers surfaces with families of hyperspheres or equidistant hyperplanes is investigated.
Journal ArticleDOI
The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space
TL;DR: The notion of lightcone Gauss maps, lightcone pedal curves and lightcone developables of spacelike curves in Minkowski 3-space was introduced in this paper.
Journal ArticleDOI
Osculating Hyperplanes and Asymptotic Directions of Codimension Two Submanifolds of Euclidean Spaces
TL;DR: In this paper, the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces were defined and necessary conditions for the convexity and sphericity of these sub-mansifolds were obtained.
References
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Book
Stable mappings and their singularities
TL;DR: In this article, the Whitney C? topology is used to classify singularities on 2-manifolds. But the Thom-Boardman invariants are not included in this classification.
Journal ArticleDOI
On singularities of submanifolds of higher dimensional Euclidean spaces
TL;DR: In this article, generalizations of principle axes are found for surfaces in E4 and the singularities generalize umbilics, and the generic indicies are computed using Thom Transversality Theorem as applied by Feldman to geometry.
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Geometry of four dimensions
TL;DR: In this paper, the authors considered a relation among points which may be called the collinear relation, and defined lines as well as planes and hyperplanes by means of this relation.
Journal ArticleDOI
The geometry of immersions. I
TL;DR: In this article, a general method for higher-order differential geometry is proposed, based on the position theory of Whitney and Thorn, and a characteristic class theory for higher order bundles having given higher order connec- tions.