Complex hyperbolic space

In mathematics, the complex hyperbolic space is an example of a negatively curved symmetrical space whose curvature - unlike the hyperbolic space - is not constant.

definition

Let be the vector space with the Hermitian form ${\displaystyle \mathbb {C} ^{n,1}}$ ${\displaystyle \mathbb {C} ^{n+1}}$

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

for . ${\displaystyle U=(u_{1},\ldots ,u_{n+1}),V=(v_{1},\ldots ,v_{n+1})}$

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

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

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

geometry

${\displaystyle \mathbb {C} H^{n}}$is isometric to the homogeneous space

${\displaystyle SU(n,1)/S(U(n)\times U(1))}$.

It is a symmetrical room of rank 1.

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

The edge at infinity is homeomorphic to . Horospheres are isometric to the Heisenberg group . ${\displaystyle \partial _{\infty }\mathbb {C} H^{n}}$ ${\displaystyle S^{2n-1}}$

Isometrics

An isometry of is called elliptic if it has a fixed point in , parabolic if it has a clear fixed point in , and loxodromic if it has two fixed points in . ${\displaystyle \mathbb {C} H^{n}}$ ${\displaystyle \mathbb {C} H^{n}}$ ${\displaystyle \partial _{\infty }\mathbb {C} H^{n}}$ ${\displaystyle \partial _{\infty }\mathbb {C} H^{n}}$

Loxodromic isometrics are represented by matrices , each with at least one eigenvalue with an amount less than or greater than 1. A loxodromic isometry is called strictly hyperbolic if it is represented by a matrix with real eigenvalues, otherwise weakly hyperbolic. ${\displaystyle A\in SU(n,1)}$ ${\displaystyle A\in SU(n,1)}$

Parabolic isometries are either unipotent; H. are represented by a matrix whose eigenvalues ​​are all 1, or ellipto-parabolic, in this case there is a unique complex geodesic on which the isometry acts as the parabolic isometry of . ${\displaystyle A\in SU(n,1)}$ ${\displaystyle \mathbb {C} H^{1}\simeq H^{2}}$

An isometric drawing is elliptical if and only if it creates a cyclic group with a compact closure. It is called regular elliptic if all eigenvalues ​​of a representative matrix are different. ${\displaystyle A\in SU(n,1)}$

Complex hyperbolic manifolds

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

Ball quotient

In algebraic geometry, complex manifolds are called ball quotients if their universal superposition is biholomorphic to . ${\displaystyle \mathbb {C} H^{n}}$

literature

• Goldman, William M .: Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. xx + 316 pp. ISBN 0-19-853793-X
• David Epstein : Complex hyperbolic geometry. Analytical and geometric aspects of hyperbolic space (Coventry / Durham, 1984), 93-111, London Math. Soc. Lecture Note Ser., 111, Cambridge Univ. Press, Cambridge, 1987.