The Horospherical Geometry of Submanifolds in Hyperbolic Space

Izumiya, Shyūichi; Pei, Donghe; Fuster, María del Carmen Romero; Takahashi, Masatomo

doi:10.1112/s0024610705006447

Cited by 32 publications

(45 citation statements)

References 28 publications

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“…As a special case of Theorem 8.6, the conditions (6), (7), (8), (9) are also equivalent. Since the suspended lightcone pedal hypersurfaces ( (5) is equivalent to the condition (9). This completes the proof.…”

Section: Proposition 94 Suppose That Nsupporting

confidence: 50%

Lightlike flat geometry of spacelike submanifolds in Lorentz–Minkowski space

Izumiya

Kasedou

2014

Int. J. Geom. Methods Mod. Phys.

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In this paper we investigate differential geometry on spacelike submanifolds in LorentzMinkowski space from the view point of contact with lightlike hyperplanes. It is called the lightlike flat geometry which has been well established for the codimension two case. In order to develop the theory for the general codimension case, we introduce the notion of codimension two spacelike canal submanifolds which is a main tool in this paper. We apply the theory of Lagrangian/Legendrian singularities to codimension two spacelike canal submanifolds and obtain the relation with the previous results on the codimension two case.

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Section: Proposition 94 Suppose That Nsupporting

confidence: 50%

Lightlike flat geometry of spacelike submanifolds in Lorentz–Minkowski space

Izumiya

Kasedou

2014

Int. J. Geom. Methods Mod. Phys.

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“…On the other hand, the horospherical geometry of higher codimension submanifolds in hyperbolic space had been developed in [11]. In this paper we continue this investigation, with the purpose of studying global properties of such submanifolds.…”

Section: Introductionmentioning

confidence: 96%

Total absolute horospherical curvature of submanifolds in hyperbolic space

Buosi

Izumiya

Ruas

2010

Advg

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Abstract. We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M, which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern-Lashof inequality for surfaces in 3-space and the horospherical Fenchel and Fary-Milnor's theorems.

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“…On the other hand, the basic notions and tools for the study of the differential geometry of hypersurfaces in hyperbolic space has recently been established in [6][7][8]. The hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space has been explicitly described and the contact of hypersurfaces with hyperhorospheres has been systematically studied as an application of singularity theory to the hyperbolic Gauss indicatrix.…”

Section: Introductionmentioning

confidence: 99%