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.

- Construction of the complex hyperbolic space
- Projective model
- Siegel model
- Group of holomorphic isometries and symmetric space
- Curvature
- Totally geodesic subspaces
- See also
- References

Complex hyperbolic spaces are also the symmetric spaces associated with the Lie groups . 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.

Let be a pseudo-Hermitian form of signature in the complex vector space . 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. It is biholomorphic to the unit ball of , as one can see by noting that a negative vector must have non zero first coordinate, and has a unique representant with first coordinate equal to 1 in the projective space. The condition when is equivalent to . The map sending the point of the unit ball of to the point of the projective space thus defines the required biholomorphism.

This model is the equivalent of the Poincaré disk model. Contrary to the real hyperbolic space, the complex projective space cannot be defined as a sheet of the hyperboloid , because the projection of this hyperboloid onto the projective model has connected fiber (the fiber being in the real case).

A Hermitian metric is defined on in the following way: if belongs to the cone , then the restriction of to the orthogonal space defines a definite positive hermitian product on this space, and because the tangent space of at the point can be naturally identified with , this defines a hermitian inner product on . As can be seen by computation, this inner product does not depend on the choice of the representant . In order to have holomorphic sectional curvature equal to -1 and not -4, one needs to renormalize this metric by a factor. This metric is a Kähler metric.

The Siegel model of complex hyperbolic space is the subset of such that

It is biholomorphic to the unit ball in via the Cayley transform

The group of holomorphic isometries of the complex hyperbolic space is the Lie group . This group acts transitively on the complex hyperbolic space, and the stabilizer of a point is isomorphic to the unitary group . The complex hyperbolic space is thus homeomorphic to the homogeneous space . The stabilizer is the maximal compact subgroup of .

As a consequence, the complex hyperbolic space is the Riemannian symmetric space ,^{ [1] } where is the pseudo-unitary group.

The group of holomorphic isometries acts transitively on the tangent complex lines of the hyperbolic complex space. 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.^{ [2] }

Furthermore, when a Hermitian manifold has constant holomorphic sectional curvature equal to , the sectional curvature of every real tangent plane is completely determined by the formula :

where is the angle between and , ie the infimum of the angles between a vector in and a vector in .^{ [2] } This angle equals 0 if and only if is a complex line, and equals if and only if is totally real. 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. Hyperbolic complex spaces can thus be seen as another high-dimensional generalization of the hyperbolic plane, less standard than the real hyperbolic spaces. A third possible generalization is the homogeneous space , which for 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 :

- a copy of a complex hyperbolic space of smaller dimension
- a copy of a real hyperbolic space of real dimension smaller than

In particular, there is no codimension 1 totally geodesic subspace of the complex hyperbolic space.

In quantum mechanics, **bra–ket notation**, or **Dirac notation**, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".

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In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In differential geometry, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold. Roughly, it states that within a fixed radius, usually called the **Margulis constant**, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

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In mathematics, and in particular gauge theory and complex geometry, a **Hermitian Yang–Mills connection** is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called **instantons**.

In mathematics, the **Segal–Bargmann space**, also known as the **Bargmann space** or **Bargmann–Fock space**, is the space of holomorphic functions *F* in *n* complex variables satisfying the square-integrability condition:

In mathematics, the **Weil–Brezin map**, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.

In algebraic geometry and differential geometry, the **nonabelian Hodge correspondence** or **Corlette–Simpson correspondence** is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold.

In complex geometry, the **Kähler identities** are a collection of identities between operators on a Kähler manifold relating the Dolbeault operators and their adjoints, contraction and wedge operators of the Kähler form, and the Laplacians of the Kähler metric. The Kähler identities combine with results of Hodge theory to produce a number of relations on de Rham and Dolbeault cohomology of compact Kähler manifolds, such as the Lefschetz hyperplane theorem, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations, and the Hodge index theorem. They are also, again combined with Hodge theory, important in proving fundamental analytical results on Kähler manifolds, such as the -lemma, the Nakano inequalities, and the Kodaira vanishing theorem.

- ↑ Arthur Besse (1987),
*Einstein manifolds*, Springer, p. 180. - 1 2 Kobayashi, Shōshichi; Nomizu, Katsumi (1996).
*Foundations of differential geometry, vol. 2*. New York: Wiley. ISBN 0-471-15733-3. OCLC 34259751.

- Goldman, William M. (1999).
*Complex hyperbolic geometry*. Oxford: Clarendon Press. p. xx + 316. ISBN 0-19-853793-X.

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