scholarly journals SINGULARITIES OF EVOLUTES OF HYPERSURFACES IN HYPERBOLIC SPACE

2004 ◽  
Vol 47 (1) ◽  
pp. 131-153 ◽  
Author(s):  
Shyuichi Izumiya ◽  
Donghe Pei ◽  
Masatomo Takahashi
Keyword(s):  
Differential Geometry ◽  
Hyperbolic Space ◽  
Subject Classification ◽  
Primary 53A35 ◽  
Lagrangian Singularities

AbstractWe study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

scholarly journals SINGULARITIES OF HYPERBOLIC GAUSS MAPS

2003 ◽  
Vol 86 (2) ◽  
pp. 485-512 ◽  
Author(s):  
SHYUICHI IZUMIYA ◽  
DONGHE PEI ◽  
TAKASI SANO
Keyword(s):  
Differential Geometry ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Singular Set ◽  
Constant Mean Curvature Surfaces ◽  
Zero Sets ◽  
Intrinsic Form ◽  
Hyperbolic Gauss Map

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

scholarly journals Invariant Surfaces under Hyperbolic Translations in Hyperbolic Space

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Mahmut Mak ◽  
Baki Karlığa
Keyword(s):  
Differential Geometry ◽  
Hyperbolic Space ◽  
Shape Operator ◽  
De Sitter ◽  
Invariant Surfaces

We consider hyperbolic rotation (G0), hyperbolic translation (G1), and horocyclic rotation (G2) groups inH3, which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups ofG0inH3. Also, we give explicit parametrization of these invariant surfaces with respect to constant hyperbolic curvature of profile curves. Finally, we obtain some corollaries for flat and minimal invariant surfaces which are associated with de Sitter and hyperbolic shape operator inH3.

Applicable Differential Geometry

1987 ◽  
Author(s):  
M. Crampin ◽  
F. A. E. Pirani
Keyword(s):  
Differential Geometry

Classical Tauberian Theorems for the (C; 1; 1) Summability Method

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Ümit Totur
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Subject Classification ◽  
Tauberian Theorems ◽  
Double Sequences ◽  
Littlewood Theorem

Abstract In this paper we generalize some classical Tauberian theorems for single sequences to double sequences. One-sided Tauberian theorem and generalized Littlewood theorem for (C; 1; 1) summability method are given as corollaries of the main results. Mathematics Subject Classification 2010: 40E05, 40G0

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