Singularities of Improper Affine Spheres and Surfaces of Constant Gaussian Curvature

Ishikawa, Goo; Machida, Yoshinori

doi:10.1142/s0129167x06003485

Cited by 53 publications

(55 citation statements)

References 18 publications

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“…and defines a left invariant distribution E on G, which induces the standard differential system E ⊂ T Z with rank 4 and with growth (4,7,10,11,12) (see [23]). In fact we can read the growth from the above The flag manifold M 9 has the canonical contact structure D M with growth (8,9), which carries a structure of 2 × 2 × 2-hyper-matrices.…”

Section: Gradations To O(4 4) and Geometric Structures On Null Flag mentioning

confidence: 99%

Geometry of D_4 conformal triality and singularities of tangent surfaces

Ishikawa¹,

Machida²,

Takahashi³

2015

jsing

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It is well known that the projective duality can be understood in the context of geometry of A n -type. In this paper, as D 4 -geometry, we construct explicitly a flag manifold, its triplefibration and differential systems which have D 4 -symmetry and conformal triality. Then we give the generic classification for singularities of the tangent surfaces to associated integral curves, which exhibits the triality. The classification is performed in terms of the classical theory on root systems combined with the singularity theory of mappings. The relations of D 4 -geometry with G 2 -geometry and B 3 -geometry are mentioned. The motivation of the tangent surface construction in D 4 -geometry is provided.

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Section: Gradations To O(4 4) and Geometric Structures On Null Flag mentioning

confidence: 99%

Geometry of D_4 conformal triality and singularities of tangent surfaces

Ishikawa¹,

Machida²,

Takahashi³

2015

jsing

Self Cite

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“…Actually, improper affine maps are given locally as a pair (Ω, f ) of a solution of (1.1), and they can be recovered in terms of their singular set. Generically, the singularities are cuspidal edges and swallowtails, (see [1,17,24,25]). …”

Section: Introductionmentioning

confidence: 99%

Ribaucour type transformations for the Hessian one equation

Martínez

Milán

Tenenblat

2015

Nonlinear Analysis: Theory, Methods & Applications

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We extend the classical theory of Ribaucour transformations to the family of improper affine maps and use it to obtain new solutions of the hessian one equation. We prove that such transformations produce complete, embedded ends of parabolic type and curves of singularities which generically are cuspidal edges. Moreover, we show that these ends and curves of singularities do no intersect. We apply Ribaucour transformations to some helicoidal improper affine maps providing new 3-parameter families with an interesting geometry and a good behavior at infinity. In particular, we construct improper affine maps, periodic in one variable, with any even number of complete embedded ends.2000 Mathematical Subject Classification: 53A15

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“…(1.2) has recently received much attention [1][2][3][4]12,14,22] which has revealed an interesting global theory for this class of surfaces.…”

Section: Introductionmentioning

confidence: 99%