Invariant eigendistributions on the tangent space of a rank one semisimple symmetric space

1984 ◽  
Vol 268 (3) ◽  
pp. 405-416 ◽  
Author(s):  
G. van Dijk
Keyword(s):  
Symmetric Space ◽  
Tangent Space ◽  
Semisimple Symmetric Space ◽  
Rank One

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Harmonic Spinors on Hyperbolic Space

1993 ◽  
Vol 36 (3) ◽  
pp. 257-262 ◽  
Author(s):  
Pierre-Yves Gaillard
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Representation Theory ◽  
Hyperbolic Space ◽  
Harmonic Functions ◽  
Short Note ◽  
Harmonic Forms ◽  
Semisimple Symmetric Space ◽  
Invariant Differential ◽  
Harmonic Spinors

AbstractThe purpose for this short note is to describe the space of harmonic spinors on hyperbolicn-spaceHn. This is a natural continuation of the study of harmonic functions onHnin [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms onHnwere considered. The connection between invariant differential equations on a Riemannian semisimple symmetric spaceG/Kand homological aspects of the representation theory ofG, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

SYMMETRIC SPACE TWO-MATRIX MODELS

1992 ◽  
Vol 07 (23) ◽  
pp. 5781-5796
Author(s):  
ARLEN ANDERSON
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Symmetric Space ◽  
Matrix Models ◽  
Explicit Expression ◽  
Lie Algebras ◽  
Spherical Function ◽  
Scaling Limits ◽  
Maximum Order ◽  
Rank One

The radial form of the partition function of a two-matrix model is formally given in terms of a spherical function for matrices representing any Euclidean symmetric space. An explicit expression is obtained by constructing the spherical function by the method of intertwining. The reduction of two-matrix models based on Lie algebras is an elementary application. A model based on the rank one symmetric space isomorphic to RN is less trivial and is treated in detail. This model may be interpreted as an Ising model on a random branched polymer. It has the unusual feature that the maximum order of criticality is different in the planar and double-scaling limits.

scholarly journals Scalar irreducibility of eigenspaces on the tangent space of a reductive symmetric space

1987 ◽  
Vol 74 (2) ◽  
pp. 292-299
Author(s):  
Henrik Schlichtkrull ◽  
Henrik Stetkær
Keyword(s):  
Symmetric Space ◽  
Tangent Space

On the complete integrability of the geodesic flow of manifolds all of whose geodesics are closed

1997 ◽  
Vol 17 (6) ◽  
pp. 1359-1370
Author(s):  
CARLOS E. DURÁN
Keyword(s):  
Finite Group ◽  
Symmetric Space ◽  
Geodesic Flow ◽  
Complete Integrability ◽  
Integrals Of Motion ◽  
Completely Integrable ◽  
Rank One ◽  
A Finite Group

We show that the geodesic flow of a metric all of whose geodesics are closed is completely integrable, with tame integrals of motion. Applications to classical examples are given; in particular, it is shown that the geodesic flow of any quotient $M/\Gamma$ of a compact, rank one symmetric space $M$ by a finite group acting freely by isometries is completely integrable by tame integrals.

scholarly journals Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

1980 ◽  
Vol 78 ◽  
pp. 1-44 ◽  
Author(s):  
Ramesh Gangolli ◽  
Garth Warner
Keyword(s):  
Symmetric Space ◽  
Zeta Function ◽  
Symmetric Spaces ◽  
Zeta Functions ◽  
Discrete Subgroup ◽  
Uniform Lattice ◽  
Rank One ◽  
Noncompact Symmetric Space

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).

Sign in / Sign up

Export Citation Format

Share Document