scholarly journals Metrical Geometry Classification of Conic Section in Hyperbolic Space Form

2012 ◽  
Vol 02 (02) ◽  
pp. 97-102
Author(s):  
幼宁 王
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Conic Section ◽  
Metrical Geometry ◽  
Hyperbolic Space Form

scholarly journals Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs

2012 ◽  
Vol 207 ◽  
pp. 95-138 ◽  
Author(s):  
Tae Hattori ◽  
Atsushi Kasue
Keyword(s):  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Space Form ◽  
Infinite Graph ◽  
Infinite Graphs ◽  
Infinite Networks ◽  
Hyperbolic Space Form

AbstractIn this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of p-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension n and it is not p-parabolic for p > n - 1, then it admits a lot of p-harmonic functions with finite Dirichlet sum of order p.

GOLDEN- AND PRODUCT-SHAPED HYPERSURFACES IN REAL SPACE FORMS

2013 ◽  
Vol 10 (04) ◽  
pp. 1320006 ◽  
Author(s):  
MIRCEA CRASMAREANU ◽  
CRISTINA-ELENA HREŢCANU ◽  
MARIAN-IOAN MUNTEANU
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Shape Operator ◽  
Real Space ◽  
Clifford Torus ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Unit Hypersphere

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS

2002 ◽  
Vol 13 (02) ◽  
pp. 209-216 ◽  
Author(s):  
JUN-MUK HWANG
Keyword(s):  
Hyperbolic Space ◽  
Complex Projective Space ◽  
Space Form ◽  
Complex Hyperbolic Space ◽  
Space Forms ◽  
Totally Geodesic ◽  
Gauss Mapping ◽  
Complex Projective ◽  
Smooth Locus ◽  
Hyperbolic Space Form

In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.

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