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.

INTEGRABILITY AND SYMMETRIC SPACES II: THE COSET SPACES

1989 ◽  
Vol 04 (03) ◽  
pp. 675-699 ◽  
Author(s):  
L. A. FERREIRA
Keyword(s):  
Symmetric Space ◽  
Poisson Bracket ◽  
Algebraic Structure ◽  
Symmetric Spaces ◽  
Sufficient Condition ◽  
Conserved Quantities ◽  
Coset Space ◽  
Canonical Variables ◽  
Coset Spaces

It is shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a Fundamental Poisson bracket Relation, and consequently charges in involution, is that it must be a symmetric space. The conditions, a Hamiltonian, or any functions of the canonical variables, has to satisfy in order to commute with these charges, are studied. It is shown that, for the case of the noncompact symmetric spaces, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities.

A rigidity theorem for manifolds without conjugate points

1998 ◽  
Vol 18 (4) ◽  
pp. 813-829 ◽  
Author(s):  
CHRISTOPHER B. CROKE ◽  
BRUCE KLEINER
Keyword(s):  
Companion Paper ◽  
Rigidity Theorem ◽  
Conjugate Points ◽  
Compactly Supported ◽  
Connected Case ◽  
Flat Metric ◽  
Simply Connected

In this paper we show that some nonsimply connected manifolds without conjugate points exhibit rigidity phenomena similar to that studied in [BGS, section I.5]. This is a companion paper to [Cr-Kl1] that deals with the simply connected case. In particular, we show that one cannot make a nontrivial, compactly supported, change to a complete flat metric without introducing conjugate points.

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 Adapted complex structures and geometric quantization

1999 ◽  
Vol 154 ◽  
pp. 171-183 ◽  
Author(s):  
Róbert Szőke
Keyword(s):  
Symmetric Space ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Symmetric Spaces ◽  
Complex Structure ◽  
Geometric Quantization ◽  
Riemannian Symmetric Space ◽  
Complex Structures ◽  
Rank One ◽  
Push Forward

AbstractA compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS.

scholarly journals Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

2013 ◽  
Vol 65 (4) ◽  
pp. 757-767 ◽  
Author(s):  
Philippe Delanoë ◽  
François Rouvière
Keyword(s):  
Symmetric Space ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Direct Proof ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Hopf Fibrations ◽  
Rank One ◽  
Simply Connected ◽  
Positively Curved

AbstractThe squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

The Focal Locus of a Riemannian 4-Symmetric Space

1988 ◽  
Vol 31 (2) ◽  
pp. 175-181
Author(s):  
J. Alfredo Jimenez
Keyword(s):  
Symmetric Space ◽  
Normal Bundle ◽  
Symmetric Spaces ◽  
Fibre Bundle ◽  
Exponential Map ◽  
Focal Points ◽  
Cut Points ◽  
Covering Map ◽  
Simply Connected

Abstractcompact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

2020 ◽  
Vol 20 (4) ◽  
pp. 499-506
Author(s):  
Julius Grüning ◽  
Ralf Köhl
Keyword(s):  
Symmetric Space ◽  
Central Extension ◽  
Hyperbolic Plane ◽  
Symmetric Spaces ◽  
Extension Theorem ◽  
Universal Property ◽  
Riemannian Symmetric Space ◽  
Differential Structure ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

AbstractBy [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

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.

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