Cartan invariant

In mathematics , the Cartan invariant is an invariant of complex-hyperbolic geometry that generalizes the double ratio of classical projective geometry , with which it can be decided in particular whether points lie in a complex or Lagrangian subspace.

definition

The infinite edge of the complex hyperbolic space is ${\displaystyle \partial _{\infty }\mathbb {C} H^{n}}$ ${\displaystyle \mathbb {C} H^{n}}$

${\displaystyle \partial _{\infty }\mathbb {C} H^{n}=\left\{X\in \mathbb {C} ^{n,1}:\langle X,X\rangle =0\right\}/\mathbb {C} }$

For each triple , Hermitean has triple product ${\displaystyle (\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right])\in \partial _{\infty }\mathbb {C} H^{n}\times \partial _{\infty }\mathbb {C} H^{n}\times \partial _{\infty }\mathbb {C} H^{n}}$

${\displaystyle \langle x_{1},x_{2},x_{3}\rangle :=\langle x_{1},x_{2}\rangle \langle x_{2},x_{3}\rangle \langle x_{3},x_{1}\rangle \in \mathbb {C} }$

because of

${\displaystyle \langle \xi _{1}x_{1},\xi _{2}x_{2},\xi _{3}x_{3}\rangle =\vert \xi _{1}\xi _{2}\xi _{3}\vert ^{2}\langle x_{1},x_{2},x_{3}\rangle }$

a well-defined argument that depends only on the equivalence classes of . So you can define ${\displaystyle x_{i}}$

${\displaystyle {\mathbb {A} }(\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right]):=arg(-\langle x_{1},x_{2},x_{3}\rangle )}$.

The function so defined

${\displaystyle {\mathbb {A} }:\partial _{\infty }\mathbb {C} H^{n}\times \partial _{\infty }\mathbb {C} H^{n}\times \partial _{\infty }\mathbb {C} H^{n}\to \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$

is called the Cartan invariant . Because the Hermitian triple product has a negative real part, the values ​​are in . ${\displaystyle \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}$

properties

The Cartan invariant is a complete invariant of triples in infinity: if for two triples of different points

${\displaystyle {\mathbb {A} }(\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right])={\mathbb {A} }(\left[y_{1}\right],\left[y_{2}\right],\left[y_{3}\right])}$

holds, then there is an isometry that converts one triple into the other. The isometry is clearly modulo isometrics that leave the plane spanned by the first triple invariant. ${\displaystyle g\in PU(n,1)}$

A triple lies in the edge of a 2-dimensional complex plane if and only if

${\displaystyle {\mathbb {A} }(\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right])=\pm {\frac {\pi }{2}}}$.

A triple lies in the edge of a Lagrangian plane if and only if

${\displaystyle {\mathbb {A} }(\left[x_{1}\right],\left[x_{2}\right],\left[x_{3}\right])=0}$.

literature

• Goldman, William M .: Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. ISBN 0-19-853793-X
• Parker, John R .; Platis, Ioannis D .: Complex hyperbolic quasi-Fuchsian groups. Geometry of Riemann surfaces, 309-355, London Math. Soc. Lecture Note Ser., 368, Cambridge Univ. Press, Cambridge, 2010.