scholarly journals Injectivity Radius and Cartan Polyhedron for Simply Connected Symmetric Spaces*

2007 ◽  
Vol 28 (6) ◽  
pp. 685-700
Author(s):  
Ling Yang
Keyword(s):  
Symmetric Spaces ◽  
Injectivity Radius ◽  
Simply Connected

POSITIVELY CURVED MANIFOLDS WITH LARGE CONJUGATE RADIUS

2013 ◽  
Vol 05 (03) ◽  
pp. 333-344 ◽  
Author(s):  
BENJAMIN SCHMIDT
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Spaces ◽  
Injectivity Radius ◽  
Curved Manifolds ◽  
Rank One ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Positively Curved

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

scholarly journals On higher covariant derivatives of the curvature tensors of Kählerian C-spaces

1983 ◽  
Vol 91 ◽  
pp. 1-18 ◽  
Author(s):  
Ryoichi Takagi
Keyword(s):  
Symmetric Spaces ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Covariant Derivatives ◽  
Higher Covariant Derivatives ◽  
Group Of Isometries ◽  
Complex Homogeneous Manifold ◽  
Curvature Tensors ◽  
Simply Connected ◽  
Derivatives Of

A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

CLOSED MINIMAL HYPERSURFACES IN COMPACT SYMMETRIC SPACES

1992 ◽  
Vol 03 (05) ◽  
pp. 629-651 ◽  
Author(s):  
CLAUDIO GORODSKI
Keyword(s):  
Symmetric Spaces ◽  
Local Geometry ◽  
Minimal Hypersurfaces ◽  
Compact Symmetric Space ◽  
Orbital Geometry ◽  
Rank One ◽  
Minimal Cones ◽  
Area Minimizing ◽  
Simply Connected ◽  
Minimal Embedding

W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.

scholarly journals Weakly complex homogeneous spaces

2014 ◽  
Vol 2014 (691) ◽  
Author(s):  
Andrei Moroianu ◽  
Uwe Semmelmann
Keyword(s):  
Tangent Bundle ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Complex Structure ◽  
Complex Spaces ◽  
Almost Complex Structures ◽  
Almost Complex ◽  
Complex Tangent ◽  
Recent Classification ◽  
Simply Connected

Abstract.We complete our recent classification (2011) of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann (1955)), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer (1986)), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.

scholarly journals The classification of simply-connected contact sub-riemannian symmetric spaces

1999 ◽  
Vol 188 (1) ◽  
pp. 65-82 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Elisha Falbel ◽  
Claudio Gorodski
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

scholarly journals Centrioles in symmetric spaces

2013 ◽  
Vol 211 ◽  
pp. 51-77
Author(s):  
Peter Quast
Keyword(s):  
Root System ◽  
Symmetric Spaces ◽  
Ambient Space ◽  
Compact Type ◽  
Geometric Properties ◽  
Simply Connected

AbstractWe describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.

Minimum and Conjugate Points in Symmetric Spaces

1962 ◽  
Vol 14 ◽  
pp. 320-328 ◽  
Author(s):  
Richard Crittenden
Keyword(s):  
Symmetric Space ◽  
Algebraic Structure ◽  
Tangent Bundle ◽  
Tangent Space ◽  
Symmetric Spaces ◽  
Conjugate Points ◽  
Minimum Cut ◽  
Cut Locus ◽  
Connected Case ◽  
Simply Connected

The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

scholarly journals Orbital decompositions of representations of non-simply connected nilpotent groups

1990 ◽  
Vol 42 (2) ◽  
pp. 293-306
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Symmetric Spaces ◽  
Integral Formula ◽  
Nilpotent Groups ◽  
Multiplicity Function ◽  
Qualitative Properties ◽  
Spectral Multiplicity ◽  
Induced Representation ◽  
Direct Integral ◽  
Orbital Parameters ◽  
Simply Connected

An orbital integral formula is proven for the direct integral decomposition of an induced representation of a connected nilpotent Lie group. Previous work required simple connectivity. An explicit description of the spectral measure and spectral multiplicity function is derived in terms of orbital parameters. It is also proven that connected (but not necessarily simply connected) exponential solvable symmetric spaces are multiplicity free. Finally, the qualitative properties of the spectral multiplicity function are examined via several illuminating examples.

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