ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Complex hyperbolic space</title>

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Complex hyperbolic space

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">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 <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a>, and it is characterised by being the only <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a> whose <a href="Holomorphic_sectional_curvature" class="mw-redirect" title="Holomorphic sectional curvature">holomorphic sectional curvature</a> is constant equal to -1. Its underlying <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> 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 <a href="CAT(k)_space" title="CAT(k) space">CAT(-1/4) space</a>.

Complex hyperbolic spaces are also the <a href="Symmetric_space" title="Symmetric space">symmetric spaces</a> associated with the <a href="Lie_group" title="Lie group">Lie groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PU(n,1)}">

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<annotation encoding="application/x-tex">{\displaystyle PU(n,1)}</annotation>

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</math><img src="./3e370d064bd5c408c7b4e84997c7b5c119f5dcaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.928ex; height:2.843ex;" alt="{\displaystyle PU(n,1)}" loading="lazy">. 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.

<div class="mw-heading mw-heading2"><h2 id="Construction_of_the_complex_hyperbolic_space">Construction of the complex hyperbolic space</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Projective_model">Projective model</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle u,v\rangle :=-u_{1}{\overline {v_{1}}}+u_{2}{\overline {v_{2}}}+\dots +u_{n+1}{\overline {v_{n+1}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \langle u,v\rangle :=-u_{1}{\overline {v_{1}}}+u_{2}{\overline {v_{2}}}+\dots +u_{n+1}{\overline {v_{n+1}}}}</annotation>

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</math><img src="./8d1ec83d608685be0b7aa54323e1b06bf356494a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.67ex; height:2.843ex;" alt="{\displaystyle \langle u,v\rangle :=-u_{1}{\overline {v_{1}}}+u_{2}{\overline {v_{2}}}+\dots +u_{n+1}{\overline {v_{n+1}}}}" loading="lazy"> be a <a href="Sesquilinear_form" title="Sesquilinear form">pseudo-Hermitian form</a> of signature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,1)}">

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<annotation encoding="application/x-tex">{\displaystyle (n,1)}</annotation>

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</math><img src="./5f8a12d0c1721c5c17ea3ba2d879264f4764402d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (n,1)}" loading="lazy"> in the complex vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./a6690b0bbf64200f073d88bdbc1bb4bc01de9b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{n+1}}" loading="lazy">. The projective model of the complex hyperbolic space is the <a href="Complex_projective_space" title="Complex projective space">projectivized space</a> of all negative vectors for this form: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle <0\}.}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle <0\}.}</annotation>

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</math><img src="./c2e55c2f77ef5ecdb270014d531c29a50f161287.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.59ex; height:3.009ex;" alt="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle <0\}.}" loading="lazy">

As an open set of the complex projective space, this space is endowed with the structure of a <a href="Complex_manifold" title="Complex manifold">complex manifold</a>. It is <a href="Biholomorphism" title="Biholomorphism">biholomorphic</a> to the unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy">, 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 <a href="Complex_projective_space" title="Complex projective space">projective space</a>. The condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \xi ,\xi \rangle <0}">

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<annotation encoding="application/x-tex">{\displaystyle \langle \xi ,\xi \rangle <0}</annotation>

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</math><img src="./d3a69c9a985c56ba4f66fedafe233fdb10a7be88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.164ex; height:2.843ex;" alt="{\displaystyle \langle \xi ,\xi \rangle <0}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi =(1,x_{1},\dots ,x_{n+1})\in \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \xi =(1,x_{1},\dots ,x_{n+1})\in \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./ed2e8f74c5c27f62c4f8446bd0fae9f752f82053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.183ex; height:3.176ex;" alt="{\displaystyle \xi =(1,x_{1},\dots ,x_{n+1})\in \mathbb {C} ^{n+1}}" loading="lazy"> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}|x_{i}|^{2}<1}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}|x_{i}|^{2}<1}</annotation>

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</math><img src="./81b97f2b7eb52617bcf24ec05389e2f8ed6d7396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.48ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}|x_{i}|^{2}<1}" loading="lazy">. The map sending the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation>

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</math><img src="./56204ada1ecf9dd5c6beddfdfd0f341cd69ff632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}" loading="lazy"> of the unit ball of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy"> to the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [1:x_{1}:\dots :x_{n}]}">

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<annotation encoding="application/x-tex">{\displaystyle [1:x_{1}:\dots :x_{n}]}</annotation>

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</math><img src="./871f312a0f9b570e02fe930eb4782addae74550b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.923ex; height:2.843ex;" alt="{\displaystyle [1:x_{1}:\dots :x_{n}]}" loading="lazy"> of the projective space thus defines the required biholomorphism.

This model is the equivalent of the <a href="Poincar%C3%A9_disk_model" title="Poincaré disk model">Poincaré disk model</a>. Unlike the real hyperbolic space, the complex projective space cannot be defined as a sheet of the hyperboloid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,x\rangle =-1}">

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<annotation encoding="application/x-tex">{\displaystyle \langle x,x\rangle =-1}</annotation>

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</math><img src="./31da52dbc90a241f9da63ce313f1ff42ca269240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.572ex; height:2.843ex;" alt="{\displaystyle \langle x,x\rangle =-1}" loading="lazy">, because the projection of this hyperboloid onto the projective model has connected fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {S} ^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {S} ^{1}}</annotation>

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</math><img src="./f091cfd6707847adccde50280b0f691f78687621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.347ex; height:2.676ex;" alt="{\displaystyle \mathbb {S} ^{1}}" loading="lazy"> (the fiber being <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation>

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</math><img src="./1b3bb21abe942aa9c0c63bae35a0c38905e1712c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }" loading="lazy"> in the real case).

A <a href="Hermitian_metric" class="mw-redirect" title="Hermitian metric">Hermitian metric</a> is defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

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</math><img src="./731170dc78b7522049c6aecf535e49817c9c80d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.227ex; height:3.009ex;" alt="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy"> in the following way: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./932fdd208979d4e6b65bc05932636bb22a1cdb06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.097ex; height:3.009ex;" alt="{\displaystyle p\in \mathbb {C} ^{n+1}}" loading="lazy"> belongs to the cone <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle p,p\rangle =-1}">

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<annotation encoding="application/x-tex">{\displaystyle \langle p,p\rangle =-1}</annotation>

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</math><img src="./8a97ec91f9259015ce8d9c89ab711252951aa6d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.251ex; height:2.843ex;" alt="{\displaystyle \langle p,p\rangle =-1}" loading="lazy">, then the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">

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<annotation encoding="application/x-tex">{\displaystyle \langle \cdot ,\cdot \rangle }</annotation>

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</math><img src="./9a50080b735975d8001c9552ac2134b49ad534c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" loading="lazy"> to the orthogonal space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {C} p)^{\perp }\subset \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle (\mathbb {C} p)^{\perp }\subset \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./59f00b5eb03c0cba467675d233c7bcbc1fc4d0c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.263ex; height:3.176ex;" alt="{\displaystyle (\mathbb {C} p)^{\perp }\subset \mathbb {C} ^{n+1}}" loading="lazy"> defines a definite positive hermitian product on this space, and because the tangent space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

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</math><img src="./731170dc78b7522049c6aecf535e49817c9c80d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.227ex; height:3.009ex;" alt="{\displaystyle \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy"> at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [p]}">

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<annotation encoding="application/x-tex">{\displaystyle [p]}</annotation>

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</math><img src="./02c90aff567de316fe4070d646975b6d8c0ab29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.463ex; height:2.843ex;" alt="{\displaystyle [p]}" loading="lazy"> can be naturally identified with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {C} p)^{\perp }}">

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<annotation encoding="application/x-tex">{\displaystyle (\mathbb {C} p)^{\perp }}</annotation>

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</math><img src="./405d73c7c5432dd3b9198dcaa79c64a8eb80901b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.168ex; height:3.176ex;" alt="{\displaystyle (\mathbb {C} p)^{\perp }}" loading="lazy">, this defines a hermitian inner product on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{[p]}\mathbb {H} _{\mathbb {C} }^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{[p]}\mathbb {H} _{\mathbb {C} }^{n}}</annotation>

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</math><img src="./2c227bbd8d369fd3e28220189997f9cfe0fad429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.558ex; height:3.176ex;" alt="{\displaystyle T_{[p]}\mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy">. As can be seen by computation, this inner product does not depend on the choice of the representative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. In order to have holomorphic sectional curvature equal to -1 and not -4, one needs to renormalize this metric by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/2}">

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</math><img src="./e308a3a46b7fdce07cc09dcab9e8d8f73e37d935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle 1/2}" loading="lazy">. This metric is a <a href="K%C3%A4hler_metric" class="mw-redirect" title="Kähler metric">Kähler metric</a>.

<div class="mw-heading mw-heading3"><h3 id="Siegel_model">Siegel model</h3></div>

The Siegel model of complex hyperbolic space is the subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (w,z)\in \mathbb {C} \times \mathbb {C} ^{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle (w,z)\in \mathbb {C} \times \mathbb {C} ^{n-1}}</annotation>

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</math><img src="./957a5d04bb751d2f759b3284f4bac1ed77714d51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.952ex; height:3.176ex;" alt="{\displaystyle (w,z)\in \mathbb {C} \times \mathbb {C} ^{n-1}}" loading="lazy"> such that

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<annotation encoding="application/x-tex">{\displaystyle i({\bar {w}}-w)>2z{\bar {z}}.}</annotation>

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</math><img src="./478c3ac6af2f6cf2b50d7aa5083dfe8372de43bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.072ex; height:2.843ex;" alt="{\displaystyle i({\bar {w}}-w)>2z{\bar {z}}.}" loading="lazy"></dd></dl>

It is biholomorphic to the unit ball in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation>

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</math><img src="./a53b4e76242764d1bca004168353c380fef25258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}" loading="lazy"> via the <a href="Cayley_transform" title="Cayley transform">Cayley transform</a>

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<annotation encoding="application/x-tex">{\displaystyle (w,z)\mapsto \left({\frac {w-i}{w+i}},{\frac {2z}{w+i}}\right).}</annotation>

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</math><img src="./2115464149ccd92d8aec9898d4370a3af5cd2a27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.985ex; height:6.176ex;" alt="{\displaystyle (w,z)\mapsto \left({\frac {w-i}{w+i}},{\frac {2z}{w+i}}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Boundary_at_infinity">Boundary at infinity</h3></div>

In the projective model, the complex hyperbolic space identifies with the complex unit ball of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, and its boundary can be defined as the boundary of the ball, which is diffeomorphic to the sphere of real dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n-1}</annotation>

</semantics>

</math><img src="./fb6ce8a02613283b1e60305814a1457335b44437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n-1}" loading="lazy">. This is equivalent to defining :

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle =0\}.}</annotation>

</semantics>

</math><img src="./3e3e850085dc91a413f0b79be3ece36c0fcdb3f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.908ex; height:3.009ex;" alt="{\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle =0\}.}" loading="lazy">

As a <a href="CAT(k)_space" title="CAT(k) space">CAT(0) space</a>, the complex hyperbolic space also has a boundary at infinity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\infty }\mathbb {H} _{\mathbb {C} }^{n}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial _{\infty }\mathbb {H} _{\mathbb {C} }^{n}}</annotation>

</semantics>

</math><img src="./bd49ef33007173be27850c002a1b98d007aa1532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.337ex; height:3.009ex;" alt="{\displaystyle \partial _{\infty }\mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy">. This boundary coincides with the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

</semantics>

</math><img src="./0308de64a24293adbbc77789ac350461f66e5f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.545ex; height:3.009ex;" alt="{\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy"> just defined.

The boundary of the complex hyperbolic space naturally carries a <a href="CR-structure" class="mw-redirect" title="CR-structure">CR structure</a>. This structure is also the standard <a href="Contact_structure" class="mw-redirect" title="Contact structure">contact structure</a> on the (odd dimensional) sphere.

<div class="mw-heading mw-heading2"><h2 id="Group_of_holomorphic_isometries_and_symmetric_space">Group of holomorphic isometries and symmetric space</h2></div>

The group of holomorphic isometries of the complex hyperbolic space is the <a href="Lie_group" title="Lie group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PU(n,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PU(n,1)}</annotation>

</semantics>

</math><img src="./3e370d064bd5c408c7b4e84997c7b5c119f5dcaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.928ex; height:2.843ex;" alt="{\displaystyle PU(n,1)}" loading="lazy">. This group acts transitively on the complex hyperbolic space, and the stabilizer of a point is isomorphic to the unitary group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(n)}</annotation>

</semantics>

</math><img src="./e9dac86141aa23bec59b25ea2c986580753b6754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.987ex; height:2.843ex;" alt="{\displaystyle U(n)}" loading="lazy">. The complex hyperbolic space is thus homeomorphic to the <a href="Homogeneous_space" title="Homogeneous space">homogeneous space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PU(n,1)/U(n)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PU(n,1)/U(n)}</annotation>

</semantics>

</math><img src="./cdf91f2079257101f49775be731951776f663837.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.077ex; height:2.843ex;" alt="{\displaystyle PU(n,1)/U(n)}" loading="lazy">. The stabilizer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(n)}</annotation>

</semantics>

</math><img src="./e9dac86141aa23bec59b25ea2c986580753b6754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.987ex; height:2.843ex;" alt="{\displaystyle U(n)}" loading="lazy"> is the <a href="Maximal_compact_subgroup" title="Maximal compact subgroup">maximal compact subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PU(n,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PU(n,1)}</annotation>

</semantics>

</math><img src="./3e370d064bd5c408c7b4e84997c7b5c119f5dcaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.928ex; height:2.843ex;" alt="{\displaystyle PU(n,1)}" loading="lazy">.

As a consequence, the complex hyperbolic space is the <a href="Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">Riemannian symmetric space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n,1)/S(U(n)\times U(1))}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle SU(n,1)/S(U(n)\times U(1))}</annotation>

</semantics>

</math><img src="./07b4f93f68b1e5ed08a918aedd059b5c4af967f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.735ex; height:2.843ex;" alt="{\displaystyle SU(n,1)/S(U(n)\times U(1))}" loading="lazy">,<a href="#cite_note-1">[1]</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle SU(n,1)}</annotation>

</semantics>

</math><img src="./9eee04043cb3ee0034f262d3044c2ecad16cac4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.682ex; height:2.843ex;" alt="{\displaystyle SU(n,1)}" loading="lazy"> is the pseudo-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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbb {D} }}=\mathbb {H} _{\mathbb {C} }^{n}\cup \partial \mathbb {H} _{\mathbb {C} }^{n}}">

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbb {D} }}=\mathbb {H} _{\mathbb {C} }^{n}\cup \partial \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

</semantics>

</math><img src="./075fb9d7aed907cc7c98ebbdb2d6dada8acc723c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.131ex; height:3.176ex;" alt="{\displaystyle {\bar {\mathbb {D} }}=\mathbb {H} _{\mathbb {C} }^{n}\cup \partial \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy">. By Brouwer's fixed point theorem, any holomorphic isometry of the complex hyperbolic space must fix at least one point in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\mathbb {D} }}}">

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {\mathbb {D} }}}</annotation>

</semantics>

</math><img src="./5ca1eba820f215964ab7c2dd4a59bb0da07ac976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.509ex;" alt="{\displaystyle {\bar {\mathbb {D} }}}" loading="lazy">. There is a classification of isometries into three types:<a href="#cite_note-2">[2]</a>

<ul><li>An isometry is said to be elliptic if it fixes a point in the complex hyperbolic space.</li>

<li>An isometry is said to be parabolic if it does not fix a point in the complex hyperbolic space and fixes a unique point in the boundary.</li>

<li>An isometry is said to be hyperbolic (or loxodromic) if it does not fix a point in the complex hyperbolic space and fixes exactly two points in the boundary.</li></ul>

The <a href="Iwasawa_decomposition" title="Iwasawa decomposition">Iwasawa decomposition</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)}</annotation>

</semantics>

</math><img src="./6e8f08e3569a57956874974b208506ac31d32779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)}" loading="lazy"> is the decomposition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)=K\times A\times N}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)=K\times A\times N}</annotation>

</semantics>

</math><img src="./c2c2a4e75bd32d9eb8ffb3bd400d7aaba0b75150.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.378ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)=K\times A\times N}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=U(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=U(n)}</annotation>

</semantics>

</math><img src="./16eef5cb631f6e8e05c5cd404d0644bb0951d07d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.151ex; height:2.843ex;" alt="{\displaystyle K=U(n)}" loading="lazy"> is the <a href="Unitary_group" title="Unitary group">unitary group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A=\mathbb {R} }</annotation>

</semantics>

</math><img src="./dd4697ca14cbcf1a7d0da6751090d999b309c741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.52ex; height:2.176ex;" alt="{\displaystyle A=\mathbb {R} }" loading="lazy"> is the additive group of real numbers and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\mathcal {H_{n}}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N={\mathcal {H_{n}}}}</annotation>

</semantics>

</math><img src="./c15fc9facc75807b8b34a4cb07220090f4c93734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.344ex; height:2.509ex;" alt="{\displaystyle N={\mathcal {H_{n}}}}" loading="lazy"> is the <a href="Heisenberg_group" title="Heisenberg group">Heisenberg group</a> of real dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2n-1}</annotation>

</semantics>

</math><img src="./fb6ce8a02613283b1e60305814a1457335b44437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n-1}" loading="lazy">. Such a decomposition depends on the choice of :

<ul><li>A point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"> in the boundary of the complex hyperbolic space (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> is then the group of <a href="Unipotent" title="Unipotent">unipotent</a> parabolic elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)}</annotation>

</semantics>

</math><img src="./6e8f08e3569a57956874974b208506ac31d32779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)}" loading="lazy"> fixing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy">)</li>

<li>An oriented <a href="Geodesic" title="Geodesic">geodesic</a> line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> going to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi }</annotation>

</semantics>

</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"> at infinity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is then the group of hyperbolic elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)}</annotation>

</semantics>

</math><img src="./6e8f08e3569a57956874974b208506ac31d32779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)}" loading="lazy"> acting as a translation along this geodesic and with no rotational part around it)</li>

<li>The choice of an origin for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy">, i.e. a unit speed parametrization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma :\mathbb {R} \to \mathbb {H} _{\mathbb {C} }^{n}}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma :\mathbb {R} \to \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

</semantics>

</math><img src="./47b4462c239de30ae284bc191cd712c492d2668e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.719ex; height:3.009ex;" alt="{\displaystyle \gamma :\mathbb {R} \to \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy"> whose image is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

</semantics>

</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is then the group of elliptic elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)}</annotation>

</semantics>

</math><img src="./6e8f08e3569a57956874974b208506ac31d32779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)}" loading="lazy"> fixing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma (0)}</annotation>

</semantics>

</math><img src="./efc6d3cbbbea2cf489b2ee4402482c5ae226f815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.234ex; height:2.843ex;" alt="{\displaystyle \gamma (0)}" loading="lazy">)</li></ul>

For any such decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PU} (n,1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {PU} (n,1)}</annotation>

</semantics>

</math><img src="./6e8f08e3569a57956874974b208506ac31d32779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.726ex; height:2.843ex;" alt="{\displaystyle \mathrm {PU} (n,1)}" loading="lazy">, the action of the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\times N}</annotation>

</semantics>

</math><img src="./39957e111b744afa0977a3859281cd3e67f3cc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.647ex; height:2.176ex;" alt="{\displaystyle A\times N}" loading="lazy"> is free and transitive, hence induces a diffeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {A} \times N\to \mathbb {H} _{\mathbb {C} }^{n}}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {A} \times N\to \mathbb {H} _{\mathbb {C} }^{n}}</annotation>

</semantics>

</math><img src="./fdd65a689636536a8eb6b73c2a12f013d042cf5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.488ex; height:3.009ex;" alt="{\displaystyle \mathrm {A} \times N\to \mathbb {H} _{\mathbb {C} }^{n}}" loading="lazy">. This diffeomorphism can be seen as a generalization of the Siegel model.

<div class="mw-heading mw-heading2"><h2 id="Curvature">Curvature</h2></div>

The group of holomorphic isometries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PU(n,1)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle PU(n,1)}</annotation>

</semantics>

</math><img src="./3e370d064bd5c408c7b4e84997c7b5c119f5dcaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.928ex; height:2.843ex;" alt="{\displaystyle PU(n,1)}" loading="lazy"> acts <a href="Group_action" title="Group action">transitively</a> on the tangent complex lines of the hyperbolic complex space. This is why this space has constant <a href="Holomorphic_sectional_curvature" class="mw-redirect" title="Holomorphic sectional curvature">holomorphic sectional curvature</a>, 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 <a href="Simply_connected_space" title="Simply connected space">simply connected</a> complete <a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a> of given constant <a href="Holomorphic_sectional_curvature" class="mw-redirect" title="Holomorphic sectional curvature">holomorphic sectional curvature</a>.<a href="#cite_note-:0-3">[3]</a>

Furthermore, when a Hermitian manifold has constant holomorphic sectional curvature equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, the sectional curvature of every real tangent plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> is completely determined by the formula :

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K(\Pi )={\frac {k}{4}}\left(1+3\cos ^{2}(\alpha (\Pi )\right)}</annotation>

</semantics>

</math><img src="./92ca678407a2194c6e8a932ca1a30734c4194efc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.944ex; height:5.343ex;" alt="{\displaystyle K(\Pi )={\frac {k}{4}}\left(1+3\cos ^{2}(\alpha (\Pi )\right)}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (\Pi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha (\Pi )}</annotation>

</semantics>

</math><img src="./8d6c309d150996aa649fdb10d1a621da0d376069.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.04ex; height:2.843ex;" alt="{\displaystyle \alpha (\Pi )}" loading="lazy"> is the angle between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J\Pi }</annotation>

</semantics>

</math><img src="./eba00085a0aa10a25cc0140f3419cb5a12f042b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.214ex; height:2.176ex;" alt="{\displaystyle J\Pi }" loading="lazy">, ie the infimum of the angles between a vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> and a vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J\Pi }</annotation>

</semantics>

</math><img src="./eba00085a0aa10a25cc0140f3419cb5a12f042b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.214ex; height:2.176ex;" alt="{\displaystyle J\Pi }" loading="lazy">.<a href="#cite_note-:0-3">[3]</a> This angle equals 0 if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> is a complex line, and equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi /2}</annotation>

</semantics>

</math><img src="./2b44e3d874a0b229fded7ffce67a0677dd5b8b67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /2}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> 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 <a href="Uniformization_theorem" title="Uniformization theorem">uniformization theorem</a>, 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 <a href="Homogeneous_space" title="Homogeneous space">homogeneous space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SL_{n}(\mathbb {R} )/SO_{n}(\mathbb {\mathbb {R} } )}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle SL_{n}(\mathbb {R} )/SO_{n}(\mathbb {\mathbb {R} } )}</annotation>

</semantics>

</math><img src="./ecc9d9e0f4bc369c29b000bf02f5ef1210736bff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.929ex; height:2.843ex;" alt="{\displaystyle SL_{n}(\mathbb {R} )/SO_{n}(\mathbb {\mathbb {R} } )}" loading="lazy">, which for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy"> again coincides with the hyperbolic plane, but becomes a symmetric space of rank greater than 1 when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

</semantics>

</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Totally_geodesic_subspaces">Totally geodesic subspaces</h2></div>

Every totally geodesic submanifold of the complex hyperbolic space of dimension n is one of the following :

<ul><li>a copy of a complex hyperbolic space of smaller dimension</li>

<li>a copy of a real hyperbolic space of real dimension smaller than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></li></ul>

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

<div class="mw-heading mw-heading2"><h2 id="Link_with_other_metrics_on_the_ball">Link with other metrics on the ball</h2></div>

<ul><li>On the unit ball, the complex hyperbolic metric coincides, up to some scalar renormalization, with the <a href="Bergman_metric" title="Bergman metric">Bergman metric</a>. This implies that every biholomorphism of the ball is actually an isometry of the complex hyperbolic metric.</li></ul>

<ul><li>The complex hyperbolic metric also coincides with the <a href="Kobayashi_metric" title="Kobayashi metric">Kobayashi metric</a>.</li></ul>

<ul><li>Up to renormalization, the complex hyperbolic metric is <a href="K%C3%A4hler%E2%80%93Einstein_metric" title="Kähler–Einstein metric">Kähler-Einstein</a>, which means that its <a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a> is a multiple of the metric.</li></ul>

<ul><li><a href="Hyperbolic_space" title="Hyperbolic space">Hyperbolic space</a></li>

<li>Quaternionic hyperbolic space</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFArthur_Besse1987" class="citation cs2"><a href="Arthur_Besse" title="Arthur Besse">Arthur Besse</a> (1987), Einstein manifolds, Springer, p. 180</cite>.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFCanoNavarreteSeade2013" class="citation book cs1">Cano, Angel; Navarrete, Juan Pablo; Seade, José (2013). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-0348-0481-3">Complex Kleinian Groups</a>. Progress in Mathematics. Vol. 303. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-0348-0481-3">10.1007/978-3-0348-0481-3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-0348-0480-6</bdi>.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <cite id="CITEREFKobayashiNomizu1996" class="citation book cs1">Kobayashi, Shōshichi; Nomizu, Katsumi (1996). Foundations of differential geometry, vol. 2. New York: Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-15733-3</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/34259751">34259751</a>.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFGoldman1999" class="citation book cs1"><a href="William_Goldman_(mathematician)" title="William Goldman (mathematician)">Goldman, William M.</a> (1999). Complex hyperbolic geometry. Oxford: Clarendon Press. p. xx + 316. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-853793-X</bdi>.</cite><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)</li></ul>

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