scholarly journals Bounded geodesics in rank-1 locally symmetric spaces

1995 ◽  
Vol 15 (5) ◽  
pp. 813-820 ◽  
Author(s):  
C. S. Aravinda ◽  
Enrico Leuzinger
Keyword(s):  
Hausdorff Dimension ◽  
Symmetric Space ◽  
Symmetric Spaces ◽  
Locally Symmetric Spaces ◽  
Tangent Sphere ◽  
Locally Symmetric Space ◽  
Riemannian Volume ◽  
Unit Vectors

AbstractLet M be a rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p ∈ M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension 1.

scholarly journals GEODESIC TUBES ON LOCALLY SYMMETRIC SPACES

1993 ◽  
Vol 24 (4) ◽  
pp. 405-416
Author(s):  
B. J. PAPANTONIOU
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Locally Symmetric Spaces ◽  
Characteristic Relation ◽  
Locally Symmetric Space

In this paper we state and prove a characteristic relation which exists, between the eigenspaces of the Ricci transformation $R(N, - )N$ acting on the orthocomplement space of $N$ in $T_mM$ where $m \in M$, $M$ being a locally symmetric space, and the Weingarten map $S_N$ of small enough geodesic tubes of $M$.

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.

scholarly journals Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift

2015 ◽  
Vol 58 (3) ◽  
pp. 632-650 ◽  
Author(s):  
Lior Silberman
Keyword(s):  
Symmetric Spaces ◽  
Differential Operators ◽  
Weak Limit ◽  
Iwasawa Decomposition ◽  
Probability Measures ◽  
Spectral Parameters ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Connected Subgroup ◽  
Invariant Differential

AbstractGiven a measureon a locally symmetric spaceobtained as a weak-* limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measureon the homogeneous spaceX= Γ\Gthat liftsand is invariant by a connected subgroupA1⊂Aof positive dimension, whereG=NAKis an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, thenis also the limit of measures associated with Hecke eigenfunctions on X. This generalizes results of the author with A.Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

η-EINSTEIN TANGENT SPHERE BUNDLES OF CONSTANT RADII

2009 ◽  
Vol 06 (06) ◽  
pp. 965-984 ◽  
Author(s):  
SUN HYANG CHUN ◽  
JEONG HYEONG PARK ◽  
KOUEI SEKIGAWA
Keyword(s):  
Symmetric Space ◽  
Einstein Manifold ◽  
Sphere Bundle ◽  
Base Manifold ◽  
Tangent Sphere Bundle ◽  
3 Dimensional ◽  
Tangent Sphere ◽  
Locally Symmetric Space ◽  
Contact Metric Structure ◽  
Standard Contact

We study the geometric properties of the base manifold for the tangent sphere bundle of radius r satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the tangent sphere bundle of the n(≥3)-dimensional locally symmetric space, equipped with the standard contact metric structure, is an η-Einstein manifold if and only if the base manifold is a space of constant sectional curvature [Formula: see text] or [Formula: see text].

scholarly journals Non-dense orbits on homogeneous spaces and applications to geometry and number theory

2021 ◽  
pp. 1-46
Author(s):  
JINPENG AN ◽  
LIFAN GUAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Locally Symmetric Spaces ◽  
Affine Map ◽  
Integer Points ◽  
Dense Orbits ◽  
Set Of Points

Abstract Let G be a Lie group, let $\Gamma \subset G$ be a discrete subgroup, let $X=G/\Gamma $ and let f be an affine map from X to itself. We give conditions on a submanifold Z of X that guarantee that the set of points $x\in X$ with f-trajectories avoiding Z is hyperplane absolute winning (a property which implies full Hausdorff dimension and is stable under countable intersections). A similar result is proved for one-parameter actions on X. This has applications in constructing exceptional geodesics on locally symmetric spaces and in non-density of the set of values of certain functions at integer points.

Tubes, Cohomology with Growth Conditions and an Application to the Theta Correspondence

1988 ◽  
Vol 40 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Stephen S. Kudla ◽  
John J. Millson
Keyword(s):  
Finite Volume ◽  
Symmetric Spaces ◽  
Integral Formula ◽  
Growth Conditions ◽  
Harmonic Forms ◽  
Theta Correspondence ◽  
Locally Symmetric Spaces ◽  
Spinor Norm ◽  
Integer Coefficients ◽  
Locally Symmetric Space

In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point of this paper is to prove an integral formula, Theorem 2.1, which will allow us to generalize the results obtained in the above papers to the finite volume case (the previous papers treated only the compact case). We then apply our integral formula to certain finite volume quotients of symmetric spaces of orthogonal groups. The main result obtained is Theorem 4.2 which is described below. We let (,) denote the bilinear form associated to a quadratic form with integer coefficients of signature (p, q). We assume that the fundamental group Γ ⊂ SO(p, q) of our locally symmetric space is the subgroup of the integral isometries of (,) congruent to the identity matrix modulo some integer N. We assume that N is chosen large enough so that Γ is neat (the multiplicative subgroup of C* generated by the eigenvalues of the elements of Γ has no torsion), Borel [2], 17.1 and that every element in Γ has spinor norm 1, Millson-Raghunathan [15], Proposition 4.1. These conditions are needed to ensure that our cycles Cx (see below) are orientable. The methods we will use apply also to unitary and quaternion unitary locally symmetric spaces, see [13].

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).

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