In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds.
The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1.
Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 (or -4 and -1, according to the choice of a normalization of the metric): in particular, it is a CAT(-1/4) space.
They constitute one of the three families of rank one symmetric spaces of noncompact type, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptional space, the Cayley plane.
The projective model of the complex hyperbolic space is the projectivized space of all negative vectors for this form:
As an open set of the complex projective space, this space is endowed with the structure of a complex manifold.
, as one can see by noting that a negative vector must have non zero first coordinate, and therefore has a unique representative with first coordinate equal to 1 in the projective space.
of the projective space thus defines the required biholomorphism.
Unlike the real hyperbolic space, the complex projective space cannot be defined as a sheet of the hyperboloid
As can be seen by computation, this inner product does not depend on the choice of the representative
In order to have holomorphic sectional curvature equal to -1 and not -4, one needs to renormalize this metric by a factor of
The Siegel model of complex hyperbolic space is the subset of
via the Cayley transform In the projective model, the complex hyperbolic space identifies with the complex unit ball of dimension
The boundary of the complex hyperbolic space naturally carries a CR structure.
This group acts transitively on the complex hyperbolic space, and the stabilizer of a point is isomorphic to the unitary group
The group of holomorphic isometries of the complex hyperbolic space also acts on the boundary of this space, and acts thus by homeomorphisms on the closed disk
By Brouwer's fixed point theorem, any holomorphic isometry of the complex hyperbolic space must fix at least one point in
There is a classification of isometries into three types:[2] The Iwasawa decomposition of
is the additive group of real numbers and
is free and transitive, hence induces a diffeomorphism
This diffeomorphism can be seen as a generalization of the Siegel model.
This is why this space has constant holomorphic sectional curvature, that can be computed to be equal to -4 (with the above normalization of the metric).
This property characterizes the hyperbolic complex space : up to isometric biholomorphism, there is only one simply connected complete Kähler manifold of given constant holomorphic sectional curvature.
[3] Furthermore, when a Hermitian manifold has constant holomorphic sectional curvature equal to
, the sectional curvature of every real tangent plane
Thus the sectional curvature of the complex hyperbolic space varies from -4 (for complex lines) to -1 (for totally real planes).
In complex dimension 1, every real plane in the tangent space is a complex line: thus the hyperbolic complex space of dimension 1 has constant curvature equal to -1, and by the uniformization theorem, it is isometric to the real hyperbolic plane.
again coincides with the hyperbolic plane, but becomes a symmetric space of rank greater than 1 when
Every totally geodesic submanifold of the complex hyperbolic space of dimension n is one of the following : In particular, there is no codimension 1 totally geodesic subspace of the complex hyperbolic space.
This differential geometry-related article is a stub.