scholarly journals SUBMANIFOLDS OF CONSTANT SCALAR CURVATURE IN A HYPERBOLIC SPACE FORM

1999 ◽  
Vol 3 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Zhong Hua Hou
Keyword(s):  
Scalar Curvature ◽  
Hyperbolic Space ◽  
Space Form ◽  
Constant Scalar Curvature ◽  
Hyperbolic Space Form

Hyperbolic submanifolds with finite type hyperbolic Gauss map

2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin
Keyword(s):  
Scalar Curvature ◽  
Finite Type ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Gauss Map ◽  
Hyperbolic Spaces ◽  
Nonzero Constant ◽  
Hyperbolic Gauss Map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

COMPACT LOCALLY CONFORMALLY FLAT RIEMANNIAN MANIFOLDS

2001 ◽  
Vol 33 (4) ◽  
pp. 459-465 ◽  
Author(s):  
QING-MING CHENG
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Space Form ◽  
Constant Scalar Curvature ◽  
Topological Classification ◽  
Conformally Flat ◽  
Riemannian Product ◽  
Locally Conformally Flat

First, we shall prove that a compact connected oriented locally conformally flat n-dimensional Riemannian manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1(c) × S1 if its Ricci curvature is nonnegative. Second, we shall give a topological classification of compact connected oriented locally conformally flat n-dimensional Riemannian manifolds with nonnegative scalar curvature r if the following inequality is satisfied: [sum ]i,jR2ij [les ] r2/(n−1), where [sum ]i,jR2ij is the squared norm of the Ricci curvature tensor.

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.

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