Quaternionic-hyperbolic space

In mathematics, the quaternionic-hyperbolic space is a negatively curved symmetrical space defined with the help of quaternions .

definition

Let be the quaternions and be the - vector space with the quaternionic-Hermitian form ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {H} ^{n,1}}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathbb {H} ^{n+1}}$

${\displaystyle \langle U,V\rangle =-u_{n+1}{\overline {v}}_{n+1}+\sum _{j=1}^{n}u_{j}{\overline {v}}_{j}}$

for . (Here the quaternionic conjugation is defined by for real numbers a, b, c, d.) ${\displaystyle U=(u_{1},\ldots ,u_{n+1}),V=(v_{1},\ldots ,v_{n+1})}$ ${\displaystyle {\overline {a+bi+cj+dk}}:=a-bi-cj-dk}$

The n-dimensional quaternionic-hyperbolic space is ${\displaystyle \mathbb {H} H^{n}}$

${\displaystyle \mathbb {H} H^{n}=\left\{X\in \mathbb {H} ^{n,1}:\langle X,X\rangle =-1\right\}}$

with the Riemannian metric induced by the Hermitian form . ${\displaystyle \langle .,.\rangle }$

Seal model

An equivalent definition can be obtained with the seal model. Here the quaternionic-Hermitian form is used , the image is viewed from under the projection onto the projective space and defined . ${\displaystyle \langle U,V\rangle ={\overline {u}}_{1}v_{n+1}+{\overline {u}}_{2}v_{2}+\ldots +{\overline {u}}_{n}v_{n}+{\overline {u}}_{n+1}v_{1}}$ ${\displaystyle V_{-}:=\left\{U\in \mathbb {H} ^{n+1}:\langle U,U\rangle <0\right\}}$ ${\displaystyle \pi :\mathbb {H} ^{n+1}\rightarrow P\mathbb {H} ^{n}}$ ${\displaystyle \mathbb {H} H^{n}:=\pi (V_{-})\subset P\mathbb {H} ^{n}}$

geometry

${\displaystyle \mathbb {H} H^{n}}$is a symmetrical space of rank 1.

The inequality applies to the sectional curvature of planes in . Planes in have section curvature , while plane has section curvature . ${\displaystyle \mathbb {H} H^{n}}$ ${\displaystyle -4\leq K\leq -1}$ ${\displaystyle \mathbb {R} H^{n}\subset \mathbb {H} H^{n}}$ ${\displaystyle -1}$ ${\displaystyle \mathbb {C} H^{1}\subset \mathbb {H} H^{1}\subset \mathbb {H} H^{n}}$ ${\displaystyle -4}$

Isometrics and quasi-isometrics

The isometry of is , this is the Lie group ${\displaystyle \mathbb {H} H^{n}}$ ${\displaystyle PSp(n,1)=Sp(n,1)/\left\{\pm 1\right\}}$ ${\displaystyle Sp(n,1)}$

${\displaystyle Sp(n,1)=\left\{A\in GL(n+1,\mathbb {H} ):\langle AU,AV\rangle =\langle U,V\rangle \forall U,V\in \mathbb {H} ^{n,1}\right\}=GL(n+1,\mathbb {H} )\cap U(2n,2)}$.

All quasi-isometrics of the have a finite distance from an isometry. ${\displaystyle \mathbb {H} H^{n}}$

Quaternionic-hyperbolic manifolds

A Riemannian manifold is called quaternionic-hyperbolic if its universal superposition is isometric to . ${\displaystyle \mathbb {H} H^{n}}$

Web links

• Jean-François Quint : An overview of Patterson-Sullivan theory pdf
• Gongopadhyay, Parsad: Classification of quaternionic hyperbolic isometries pdf

swell

1. ^ Inkang Kim , John R. Parker : Geometry of quaternionic hyperbolic manifolds . In: Cambridge Philosophical Society : Mathematical Proceedings , 135 (2003), no. 2, 291-320. ISSN  0305-0041 pdf
2. ↑ Pierre Pansu : Métriques de Carnot-Carathéodory et des espaces quasiisométries symétriques de rang un . In: Annals of Mathematics , (2) 129 (1989), no. 1, 1-60. ISSN  0003-486X pdf