scholarly journals Eigenspaces of the Laplace-Beltrami operator on a hyperboloid

1980 ◽  
Vol 79 ◽  
pp. 151-185 ◽  
Author(s):  
Jiro Sekiguchi
Keyword(s):  
Differential Equations ◽  
Symmetric Space ◽  
Differential Operators ◽  
Beltrami Operator ◽  
Poisson Integral ◽  
Riemannian Symmetric Space ◽  
Systems Of Differential Equations ◽  
Rank One ◽  
Joint Eigenfunctions ◽  
Invariant Differential

Ever since S. Helgason [4] showed that any eigenfunction of the Laplace-Beltrami operator on the unit disk is represented by the Poisson integral of a hyperfunction on the unit circle, much interest has been arisen to the study of the Poisson integral representation of joint eigenfunctions of all invariant differential operators on a symmetric space X. In particular, his original idea of expanding eigenfunctions into K-finite functions has proved to be generalizable up to the case where X is a Riemannian symmetric space of rank one (cf. [4], [5], [11]). Presently, extension to arbitrary rank has been completed by quite a different formalism which views the present problem as a boundary-value problem for the differential equations. It should be recalled that along this line of approach a general theory of the systems of differential equations with regular singularities was successfully established by Kashiwara-Oshima (cf. [6], [7]).

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

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.

Eigenfunctions of Invariant Differential Operators on a Symmetric Space

1978 ◽  
Vol 107 (1) ◽  
pp. 1 ◽  
Author(s):  
M. Kashiwara ◽  
A. Kowata ◽  
K. Minemura ◽  
K. Okamoto ◽  
T. Oshima ◽  
...  
Keyword(s):  
Symmetric Space ◽  
Differential Operators ◽  
Invariant Differential Operators ◽  
Invariant Differential

scholarly journals Distribution of length spectrum of circles on a complex hyperbolic space

1999 ◽  
Vol 153 ◽  
pp. 119-140 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Symmetric Space ◽  
Hyperbolic Space ◽  
Real Line ◽  
Isometry Group ◽  
Space Form ◽  
Geodesic Curvature ◽  
Real Space ◽  
Complex Hyperbolic Space ◽  
Riemannian Symmetric Space ◽  
Rank One

AbstractIt is well-known that all geodesics on a Riemannian symmetric space of rank one are congruent each other under the action of isometry group. Being concerned with circles, we also know that two closed circles in a real space form are congruent if and only if they have the same length. In this paper we study how prime periods of circles on a complex hyperbolic space are distributed on a real line and show that even if two circles have the same length and the same geodesic curvature they are not necessarily congruent each other.

Sign in / Sign up

Export Citation Format

Share Document