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.
a well-defined argument that depends only on the equivalence classes of . So you can define
.
The function so defined
is called the Cartan invariant . Because the Hermitian triple product has a negative real part, the values are in .
properties
The Cartan invariant is a complete invariant of triples in infinity: if for two triples of different points
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.
A triple lies in the edge of a 2-dimensional complex plane if and only if
.
A triple lies in the edge of a Lagrangian plane if and only if
.
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.