The Moduli Space of Complex Geodesics in the Complex Hyperbolic Plane

2010 ◽  
Vol 22 (2) ◽  
pp. 295-319 ◽  
Author(s):  
Heleno Cunha ◽  
Francisco Dutenhefner ◽  
Nikolay Gusevskii ◽  
Rafael Santos Thebaldi
Keyword(s):  
Moduli Space ◽  
Hyperbolic Plane ◽  
Complex Hyperbolic Plane

SPHERICAL FUNCTIONS, THE COMPLEX HYPERBOLIC PLANE AND THE HYPERGEOMETRIC OPERATOR

2006 ◽  
Vol 17 (10) ◽  
pp. 1151-1173 ◽  
Author(s):  
P. ROMÁN ◽  
J. TIRAO
Keyword(s):  
Hypergeometric Function ◽  
Hyperbolic Plane ◽  
Casimir Operator ◽  
Differential Operators ◽  
Real Variable ◽  
Spherical Functions ◽  
Matrix Coefficients ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Complex Hyperbolic Plane

In this paper, we determine all irreducible spherical functions Φ of any K-type associated to the dual Hermitian symmetric pairs (G, K) = ( SU (3), U (2)) and ( SU (2,1), U (2)). This is accomplished by associating to Φ a vector valued function H = H(u) of a real variable u, analytic at u = 0, which is a simultaneous eigenfunction of two second order differential operators with matrix coefficients. One of them comes from the Casimir operator of G and we prove that it is conjugated to a hypergeometric operator, allowing us to express the function H in terms of a matrix valued hypergeometric function. For the compact pair ( SU (3), U (2)), this project was started in [4].

scholarly journals Cross-ratio Dynamics on Ideal Polygons

2020 ◽  
Author(s):  
Maxim Arnold ◽  
Dmitry Fuchs ◽  
Ivan Izmestiev ◽  
Serge Tabachnikov
Keyword(s):  
Hyperbolic Space ◽  
Moduli Space ◽  
Hyperbolic Plane ◽  
Projective Line ◽  
Cross Ratio ◽  
Poisson Structures ◽  
Completely Integrable ◽  
The Cross

Abstract Two ideal polygons, $(p_1,\ldots ,p_n)$ and $(q_1,\ldots ,q_n)$, in the hyperbolic plane or in hyperbolic space are said to be $\alpha $-related if the cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha $ for all $i$ (the vertices lie on the projective line, real or complex, respectively). For example, if $\alpha = -1$, the respective sides of the two polygons are orthogonal. This relation extends to twisted ideal polygons, that is, polygons with monodromy, and it descends to the moduli space of Möbius-equivalent polygons. We prove that this relation, which is generically a 2-2 map, is completely integrable in the sense of Liouville. We describe integrals and invariant Poisson structures and show that these relations, with different values of the constants $\alpha $, commute, in an appropriate sense. We investigate the case of small-gons and describe the exceptional ideal pentagons and hexagons that possess infinitely many $\alpha $-related polygons.

The geometry of complex hyperbolic packs

2009 ◽  
Vol 147 (1) ◽  
pp. 205-234 ◽  
Author(s):  
IOANNIS D. PLATIS
Keyword(s):  
Hyperbolic Plane ◽  
Complex Hyperbolic Plane

AbstractComplex hyperbolic packs are hypersurfaces of complex hyperbolic planeH2ℂwhich may be considered as dual to the well known bisectors. In this paper we study the geometric aspects associated to packs.

Real Hypersurfaces in ℂP 2 and ℂH 2 with Cyclic Parallel ∗-Ricci Tensor

2021 ◽  
Vol 58 (3) ◽  
pp. 308-318
Author(s):  
Yaning Wang ◽  
Wenjie Wang
Keyword(s):  
Ricci Tensor ◽  
Hyperbolic Plane ◽  
Real Hypersurface ◽  
Three Dimensional ◽  
Real Hypersurfaces ◽  
Complex Projective Plane ◽  
Cyclic Parallel Ricci Tensor ◽  
Complex Projective ◽  
Parallel Ricci Tensor ◽  
Complex Hyperbolic Plane

In this paper, we prove that the ∗-Ricci tensor of a real hypersurface in complex projective plane ℂP 2 or complex hyperbolic plane ℂH 2 is cyclic parallel if and only if the hypersurface is of type (A). We find some three-dimensional real hypersurfaces having non-vanishing and non-parallel ∗-Ricci tensors which are cyclic parallel.

scholarly journals Traces, Cross-Ratios and 2-Generator Subgroups of SU(2, 1)

2009 ◽  
Vol 61 (6) ◽  
pp. 1407-1436 ◽  
Author(s):  
Pierre Will
Keyword(s):  
Hyperbolic Plane ◽  
Complex Hyperbolic Plane

Abstract In this work, we investigate how to decompose a pair (A, B) of loxodromic isometries of the complex hyperbolic plane H2𝕔 under the form A = I1I2 and B = I3I2, where the Ik's are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group ‹A, B›.

scholarly journals Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

2021 ◽  
pp. 1-34
Author(s):  
LAURENT DUFLOUX ◽  
VILLE SUOMALA
Keyword(s):  
Hausdorff Dimension ◽  
Heisenberg Group ◽  
Hyperbolic Plane ◽  
Strong Version ◽  
Empty Interior ◽  
Vertical Projection ◽  
Euclidean Metric ◽  
The One ◽  
One Point Compactification ◽  
Complex Hyperbolic Plane

Abstract We study projectional properties of Poisson cut-out sets E in non-Euclidean spaces. In the first Heisenbeg group \[\mathbb{H} = \mathbb{C} \times \mathbb{R}\] , endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection \[\pi (E)\] (projection along the center of \[\mathbb{H}\] ) almost surely equals \[\min \{ 2,{\dim _\operatorname{H} }(E)\} \] and that \[\pi (E)\] has non-empty interior if \[{\dim _{\text{H}}}(E) > 2\] . As a corollary, this allows us to determine the Hausdorff dimension of E with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension \[{\dim _{\text{H}}}(E)\] . We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere \[{{\text{S}}^3}\] endowed with the visual metric d obtained by identifying \[{{\text{S}}^3}\] with the boundary of the complex hyperbolic plane. In \[{{\text{S}}^3}\] , we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in \[{{\text{S}}^3}\] satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions.

scholarly journals Polar actions on the complex hyperbolic plane

2012 ◽  
Vol 43 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Jürgen Berndt ◽  
José Carlos Díaz-Ramos
Keyword(s):  
Hyperbolic Plane ◽  
Polar Actions ◽  
Complex Hyperbolic Plane
Sign in / Sign up

Export Citation Format

Share Document