Orbits of a stationary subgroup on a pseudo-Riemannian symmetric space of rank one

1983 ◽  
Vol 38 (5) ◽  
pp. 158-159
Author(s):  
V F Molchanov
Keyword(s):  
Symmetric Space ◽  
Riemannian Symmetric Space ◽  
Stationary Subgroup ◽  
Rank One

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.

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]).

scholarly journals On the least positive eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space

1980 ◽  
Vol 78 ◽  
pp. 137-152 ◽  
Author(s):  
Hajime Urakawa
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Lie Group ◽  
Discrete Subgroup ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Identity Component ◽  
Rank One ◽  
Quotient Manifold

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

scholarly journals EXTERIOR DIFFERENTIAL FORMS ON RIEMANNIAN SYMMETRIC SPACES

2017 ◽  
pp. 49-53
Author(s):  
Irina Alexandrova ◽  
Irina Alexandrova ◽  
Sergey Stepanov ◽  
Sergey Stepanov ◽  
Irina Tsyganok ◽  
...  
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Differential Forms ◽  
Conformal Killing ◽  
Riemannian Symmetric Space ◽  
Vanishing Theorems ◽  
Harmonic Forms ◽  
Exterior Differential ◽  
Rank One ◽  
Riemannian Symmetric Spaces

In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

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 An Anosov action on the bundle of Weyl chambers

1985 ◽  
Vol 5 (4) ◽  
pp. 587-593 ◽  
Author(s):  
Hans-Christoph Im Hof
Keyword(s):  
Symmetric Space ◽  
Geodesic Flow ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Rank One

AbstractWe introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.

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.

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