Splittings and Cr-structures for manifolds with nonpositive sectional curvature

2001 ◽  
Vol 144 (1) ◽  
pp. 139-167 ◽  
Author(s):  
Jianguo Cao ◽  
Jeff Cheeger ◽  
Xiaochun Rong
Keyword(s):  
Sectional Curvature ◽  
Nonpositive Sectional Curvature ◽  
Cr Structures

scholarly journals The double cover relative to a convex domain and the relative isoperimetric inequality

2006 ◽  
Vol 80 (3) ◽  
pp. 375-382 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Convex Domain ◽  
Double Cover ◽  
Nonpositive Sectional Curvature ◽  
Relative Isoperimetric Inequality ◽  
Simply Connected

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

scholarly journals Para-CR structures on almost paracontact metric manifolds

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Joanna Wełyczko
Keyword(s):  
Sectional Curvature ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Cr Manifolds ◽  
Cosymplectic Manifolds ◽  
Cr Structures ◽  
Necessary And Sufficient

AbstractAlmost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and sufficient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that normal almost paracontact metric manifolds are para-CR. We establish necessary and sufficient conditions for paracontact metric manifolds as well as for almost para-cosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greater than 3 must be para-Sasakian and its curvature equal to -1. The last assertion does not hold in dimension 3. We show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to -1. New classes of examples of para-CR manifolds are established.

GEOMETRY OF HOMOGENEOUS HADAMARD MANIFOLDS

1991 ◽  
Vol 02 (02) ◽  
pp. 223-234 ◽  
Author(s):  
THOMAS H. WOLTER
Keyword(s):  
Sectional Curvature ◽  
Algebraic Structure ◽  
Symmetric Spaces ◽  
Hadamard Manifolds ◽  
Homogeneous Manifolds ◽  
Nonpositive Sectional Curvature

Using result of R. Azencott and E. Wilson on the algebraic structure of homogeneous manifolds of nonpositive sectional curvature, we discuss the geometry of these manifolds and give several conditions how to distinguish the symmetric spaces among them.

scholarly journals The balance between existence and nonexistence theorems in differential geometry

2001 ◽  
Vol 32 (1) ◽  
pp. 61-88
Author(s):  
Shihsuh Walter Wei
Keyword(s):  
Differential Geometry ◽  
Sectional Curvature ◽  
Harmonic Maps ◽  
Minimal Submanifolds ◽  
Nonlinear Pde ◽  
Minimal Hypersurfaces ◽  
Nonpositive Sectional Curvature ◽  
Existence And Nonexistence ◽  
Complete Manifolds ◽  
Topological Theorem

We discuss the delicate balance between existence and nonexistence theorems in differential geometry. Studying their interplay yields some information about $ p $-harmonic maps, $ p $-SSU manifolds, geometric $ k_p $-connected manifolds, minimal hypersurfaces and Gauss maps, and manifolds admitting essential positive supersolutions of certain nonlinear PDE. As an application of the theory developed, we obtain a topological theorem for minimal submanifolds in complete manifolds with nonpositive sectional curvature.

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