Horospherical flat surfaces in Hyperbolic 3-space

Izumiya, Shyūichi; Saji, Kentaro; Takahashi, Masatomo

doi:10.2969/jmsj/06230789

Cited by 60 publications

(94 citation statements)

References 29 publications

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“…Since c 2 = c 1 − c 4 = 0 hold, the surface X h is horo-flat in the sense of [22]. Moreover if we assume c 3 (t) = 0 for any t, then the singular value of X h is a 0 (t).…”

Section: By the Above Construction (Xmentioning

confidence: 99%

“…In [22], we investigated "horo-flat" horospherical surfaces in H 3 + (−1). It is linear Weingarten surfaces of non-Bryant type, we considered them as surfaces whose hyperbolic Gauss map degenerates to a curve in the de Sitter space (see [22,Section 4]). This means that a horo-flat horospherical surface is the dual surface of a curve in the de Sitter space.…”

Section: Linear Weingarten Surfaces Of Bryant Type and Bianchi Typementioning

confidence: 99%

“…In [22], we introduced these surfaces X h and X h by the same construction as the above and investigated the geomeotric properties and singularities of them. It has been shown in [22] …”

Section: T)mentioning

confidence: 99%

“…Especially, we consider the case when the Legendrian dual is a curve in a pseudo-sphere. In [22] we have studied a surface in Hyperbolic space whose lightcone dual is a curve.In this case the suface is called a horo-flat surface. Moreover, such surfaces are one-parameter families of horo-cycles.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we call it a horo-flat horocyclic surface. Horo-flat surfaces are "flat"surfaces in the sense of a new geometry in Hyperbolic space [5,6,17,18,19,22] which is called "Horospherical Geometry".…”

Section: Introductionmentioning

confidence: 99%

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The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Izumiya¹,

Saji²

2010

J. Sing.

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Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-Minkowski 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the pseudo-sphere and its dual surface. Flatness of a surface is defined by the degeneracy of the dual surface similar to the case for the Gauss map of a flat surface in the Euclidean space. We study singularities of these flat surfaces and dualities of singularities.

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“…Since c 2 = c 1 − c 4 = 0 hold, the surface X h is horo-flat in the sense of [22]. Moreover if we assume c 3 (t) = 0 for any t, then the singular value of X h is a 0 (t).…”

Section: By the Above Construction (Xmentioning

confidence: 99%

Section: Linear Weingarten Surfaces Of Bryant Type and Bianchi Typementioning

confidence: 99%

“…In [22], we introduced these surfaces X h and X h by the same construction as the above and investigated the geomeotric properties and singularities of them. It has been shown in [22] …”

Section: T)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Izumiya¹,

Saji²

2010

J. Sing.

Self Cite

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Orientability of linear Weingarten surfaces, spacelike CMC‐1 surfaces and maximal surfaces

Kokubu

Umehara

2011

Mathematische Nachrichten

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We prove several topological properties of linear Weingarten surfaces of Bryant type, as wave fronts in hyperbolic 3-space. For example, we show the orientability of such surfaces, and also co-orientability when they are not flat. Moreover, we show an explicit formula of the non-holomorphic hyperbolic Gauss map via another hyperbolic Gauss map which is holomorphic. Using this, we show the orientability and co-orientability of CMC-1 faces (i.e., constant mean curvature one surfaces with admissible singular points) in de Sitter 3-space.(CMC-1 faces might not be wave fronts in general, but belong to a class of linear Weingarten surfaces with singular points.) Since both linear Weingarten fronts and CMC-1 faces may have singular points, orientability and co-orientability are both nontrivial properties. Furthermore, we show that the zig-zag representation of the fundamental group of a linear Weingarten surface of Bryant type is trivial. We also remark on some properties of non-orientable maximal surfaces in Lorentz-Minkowski 3-space, comparing the corresponding properties of CMC-1 faces in de Sitter 3-space.

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Dualities of Differential Geometric Invariants on Cuspidal Edges on Flat Fronts in the Hyperbolic Space and the de Sitter Space

Saji

Teramoto

2020

Mediterr. J. Math.

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Horospherical flat surfaces in Hyperbolic 3-space

Cited by 60 publications

References 29 publications

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces

Orientability of linear Weingarten surfaces, spacelike CMC‐1 surfaces and maximal surfaces

Dualities of Differential Geometric Invariants on Cuspidal Edges on Flat Fronts in the Hyperbolic Space and the de Sitter Space

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