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Complex hyperbolic space
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<h2 id="construction-of-the-complex-hyperbolic-space">Construction of the complex hyperbolic space</h2> <h3>Projective model</h3> <p>Let ⟨ u , v ⟩ := − u 1 v 1 ¯ + u 2 v 2 ¯ + ⋯ + u n + 1 v n + 1 ¯ {\displaystyle \langle u,v\rangle :=-u_{1}{\overline {v_{1}}}+u_{2}{\overline {v_{2}}}+\dots +u_{n+1}{\overline {v_{n+1}}}} be a <a href="/facts/Sesquilinear_form/Kp6qUA8v">pseudo-Hermitian form</a> of signature ( n , 1 ) {\displaystyle (n,1)} in the complex vector space C n + 1 {\displaystyle \mathbb {C} ^{n+1}} . The projective model of the complex hyperbolic space is the <a href="/facts/Complex_projective_space/QmgneIzS">projectivized space</a> of all negative vectors for this form: H C n = { [ ξ ] ∈ C P n | ⟨ ξ , ξ ⟩ < 0 } . {\displaystyle \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle <0\}.} </p><p>As an open set of the complex projective space, this space is endowed with the structure of a <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a>. It is <a href="/facts/Biholomorphism/3g5kL1RD">biholomorphic</a> to the unit ball of C n {\displaystyle \mathbb {C} ^{n}} , 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="/facts/Complex_projective_space/QmgneIzS">projective space</a>. The condition ⟨ ξ , ξ ⟩ < 0 {\displaystyle \langle \xi ,\xi \rangle <0} when ξ = ( 1 , x 1 , … , x n + 1 ) ∈ C n + 1 {\displaystyle \xi =(1,x_{1},\dots ,x_{n+1})\in \mathbb {C} ^{n+1}} is equivalent to ∑ i = 1 n | x i | 2 < 1 {\displaystyle \sum _{i=1}^{n}|x_{i}|^{2}<1} . The map sending the point ( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} of the unit ball of C n {\displaystyle \mathbb {C} ^{n}} to the point [ 1 : x 1 : ⋯ : x n ] {\displaystyle [1:x_{1}:\dots :x_{n}]} of the projective space thus defines the required biholomorphism. </p><p>This model is the equivalent of the <a href="/facts/Poincar%25C3%25A9_disk_model/6nCWjbRL">Poincaré disk model</a>. Unlike the real hyperbolic space, the complex projective space cannot be defined as a sheet of the hyperboloid ⟨ x , x ⟩ = − 1 {\displaystyle \langle x,x\rangle =-1} , because the projection of this hyperboloid onto the projective model has connected fiber S 1 {\displaystyle \mathbb {S} ^{1}} (the fiber being Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } in the real case). </p><p>A <a href="/facts/Hermitian_metric/69fYok0x">Hermitian metric</a> is defined on H C n {\displaystyle \mathbb {H} _{\mathbb {C} }^{n}} in the following way: if p ∈ C n + 1 {\displaystyle p\in \mathbb {C} ^{n+1}} belongs to the cone ⟨ p , p ⟩ = − 1 {\displaystyle \langle p,p\rangle =-1} , then the restriction of ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } to the orthogonal space ( C p ) ⊥ ⊂ C n + 1 {\displaystyle (\mathbb {C} p)^{\perp }\subset \mathbb {C} ^{n+1}} defines a definite positive hermitian product on this space, and because the tangent space of H C n {\displaystyle \mathbb {H} _{\mathbb {C} }^{n}} at the point [ p ] {\displaystyle [p]} can be naturally identified with ( C p ) ⊥ {\displaystyle (\mathbb {C} p)^{\perp }} , this defines a hermitian inner product on T [ p ] H C n {\displaystyle T_{[p]}\mathbb {H} _{\mathbb {C} }^{n}} . As can be seen by computation, this inner product does not depend on the choice of the representative p {\displaystyle p} . In order to have holomorphic sectional curvature equal to -1 and not -4, one needs to renormalize this metric by a factor of 1 / 2 {\displaystyle 1/2} . This metric is a <a href="/facts/K%25C3%25A4hler_metric/JFPM6Vxg">Kähler metric</a>. </p> <h3>Siegel model</h3> <p>The Siegel model of complex hyperbolic space is the subset of ( w , z ) ∈ C × C n − 1 {\displaystyle (w,z)\in \mathbb {C} \times \mathbb {C} ^{n-1}} such that </p> i ( w ¯ − w ) > 2 z z ¯ . {\displaystyle i({\bar {w}}-w)>2z{\bar {z}}.} <p>It is biholomorphic to the unit ball in C n {\displaystyle \mathbb {C} ^{n}} via the <a href="/facts/Cayley_transform/2jbbGdA9">Cayley transform</a> </p> ( w , z ) ↦ ( w − i w + i , 2 z w + i ) . {\displaystyle (w,z)\mapsto \left({\frac {w-i}{w+i}},{\frac {2z}{w+i}}\right).} <h3>Boundary at infinity</h3> <p>In the projective model, the complex hyperbolic space identifies with the complex unit ball of dimension n {\displaystyle n} , and its boundary can be defined as the boundary of the ball, which is diffeomorphic to the sphere of real dimension 2 n − 1 {\displaystyle 2n-1} . This is equivalent to defining : ∂ H C n = { [ ξ ] ∈ C P n | ⟨ ξ , ξ ⟩ = 0 } . {\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}=\{[\xi ]\in \mathbb {CP} ^{n}|\langle \xi ,\xi \rangle =0\}.} </p><p>As a <a href="/facts/CAT(k)_space/fmnJjYsN">CAT(0) space</a>, the complex hyperbolic space also has a boundary at infinity ∂ ∞ H C n {\displaystyle \partial _{\infty }\mathbb {H} _{\mathbb {C} }^{n}} . This boundary coincides with the boundary ∂ H C n {\displaystyle \partial \mathbb {H} _{\mathbb {C} }^{n}} just defined. </p><p>The boundary of the complex hyperbolic space naturally carries a <a href="/facts/CR-structure/ECXTscxj">CR structure</a>. This structure is also the standard <a href="/facts/Contact_structure/r3IswjXw">contact structure</a> on the (odd dimensional) sphere. </p> <h2 id="group-of-holomorphic-isometries-and-symmetric-space">Group of holomorphic isometries and symmetric space</h2> <p>The group of holomorphic isometries of the complex hyperbolic space is the <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> P U ( n , 1 ) {\displaystyle PU(n,1)} . This group acts transitively on the complex hyperbolic space, and the stabilizer of a point is isomorphic to the unitary group U ( n ) {\displaystyle U(n)} . The complex hyperbolic space is thus homeomorphic to the <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> P U ( n , 1 ) / U ( n ) {\displaystyle PU(n,1)/U(n)} . The stabilizer U ( n ) {\displaystyle U(n)} is the <a href="/facts/Maximal_compact_subgroup/8EA7XRcu">maximal compact subgroup</a> of P U ( n , 1 ) {\displaystyle PU(n,1)} . </p><p>As a consequence, the complex hyperbolic space is the <a href="/facts/Riemannian_symmetric_space/AWYuHfNp">Riemannian symmetric space</a> S U ( n , 1 ) / S ( U ( n ) × U ( 1 ) ) {\displaystyle SU(n,1)/S(U(n)\times U(1))} ,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> where S U ( n , 1 ) {\displaystyle SU(n,1)} is the pseudo-unitary group. </p><p>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 D ¯ = H C n ∪ ∂ H C n {\displaystyle {\bar {\mathbb {D} }}=\mathbb {H} _{\mathbb {C} }^{n}\cup \partial \mathbb {H} _{\mathbb {C} }^{n}} . By Brouwer's fixed point theorem, any holomorphic isometry of the complex hyperbolic space must fix at least one point in D ¯ {\displaystyle {\bar {\mathbb {D} }}} . There is a classification of isometries into three types:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> <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> <p>The <a href="/facts/Iwasawa_decomposition/g0EWM61r">Iwasawa decomposition</a> of P U ( n , 1 ) {\displaystyle \mathrm {PU} (n,1)} is the decomposition P U ( n , 1 ) = K × A × N {\displaystyle \mathrm {PU} (n,1)=K\times A\times N} , where K = U ( n ) {\displaystyle K=U(n)} is the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a>, A = R {\displaystyle A=\mathbb {R} } is the additive group of real numbers and N = H n {\displaystyle N={\mathcal {H_{n}}}} is the <a href="/facts/Heisenberg_group/rsqq1426">Heisenberg group</a> of real dimension 2 n − 1 {\displaystyle 2n-1} . Such a decomposition depends on the choice of : </p> <ul><li>A point ξ {\displaystyle \xi } in the boundary of the complex hyperbolic space ( N {\displaystyle N} is then the group of <a href="/facts/Unipotent/xnJlzBR9">unipotent</a> parabolic elements of P U ( n , 1 ) {\displaystyle \mathrm {PU} (n,1)} fixing ξ {\displaystyle \xi } )</li> <li>An oriented <a href="/facts/Geodesic/fbWR6l80">geodesic</a> line ℓ {\displaystyle \ell } going to ξ {\displaystyle \xi } at infinity ( A {\displaystyle A} is then the group of hyperbolic elements of P U ( n , 1 ) {\displaystyle \mathrm {PU} (n,1)} acting as a translation along this geodesic and with no rotational part around it)</li> <li>The choice of an origin for ℓ {\displaystyle \ell } , i.e. a unit speed parametrization γ : R → H C n {\displaystyle \gamma :\mathbb {R} \to \mathbb {H} _{\mathbb {C} }^{n}} whose image is ℓ {\displaystyle \ell } ( K {\displaystyle K} is then the group of elliptic elements of P U ( n , 1 ) {\displaystyle \mathrm {PU} (n,1)} fixing γ ( 0 ) {\displaystyle \gamma (0)} )</li></ul> <p>For any such decomposition of P U ( n , 1 ) {\displaystyle \mathrm {PU} (n,1)} , the action of the subgroup A × N {\displaystyle A\times N} is free and transitive, hence induces a diffeomorphism A × N → H C n {\displaystyle \mathrm {A} \times N\to \mathbb {H} _{\mathbb {C} }^{n}} . This diffeomorphism can be seen as a generalization of the Siegel model. </p> <h2 id="curvature">Curvature</h2> <p>The group of holomorphic isometries P U ( n , 1 ) {\displaystyle PU(n,1)} acts <a href="/facts/Group_action/Vh9VtUUw">transitively</a> on the tangent complex lines of the hyperbolic complex space. This is why this space has constant <a href="/facts/Holomorphic_sectional_curvature/JFPM6Vxg">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="/facts/Simply_connected_space/xFlta63J">simply connected</a> complete <a href="/facts/K%25C3%25A4hler_manifold/JFPM6Vxg">Kähler manifold</a> of given constant <a href="/facts/Holomorphic_sectional_curvature/JFPM6Vxg">holomorphic sectional curvature</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Furthermore, when a Hermitian manifold has constant holomorphic sectional curvature equal to k {\displaystyle k} , the sectional curvature of every real tangent plane Π {\displaystyle \Pi } is completely determined by the formula : </p><p> K ( Π ) = k 4 ( 1 + 3 cos 2 ( α ( Π ) ) {\displaystyle K(\Pi )={\frac {k}{4}}\left(1+3\cos ^{2}(\alpha (\Pi )\right)} </p><p>where α ( Π ) {\displaystyle \alpha (\Pi )} is the angle between Π {\displaystyle \Pi } and J Π {\displaystyle J\Pi } , ie the infimum of the angles between a vector in Π {\displaystyle \Pi } and a vector in J Π {\displaystyle J\Pi } .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> This angle equals 0 if and only if Π {\displaystyle \Pi } is a complex line, and equals π / 2 {\displaystyle \pi /2} if and only if Π {\displaystyle \Pi } is totally real. Thus the sectional curvature of the complex hyperbolic space varies from -4 (for complex lines) to -1 (for totally real planes). </p><p>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="/facts/Uniformization_theorem/mEa8a6IQ">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="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> S L n ( R ) / S O n ( R ) {\displaystyle SL_{n}(\mathbb {R} )/SO_{n}(\mathbb {\mathbb {R} } )} , which for n = 2 {\displaystyle n=2} again coincides with the hyperbolic plane, but becomes a symmetric space of rank greater than 1 when n ≥ 3 {\displaystyle n\geq 3} . </p> <h2 id="totally-geodesic-subspaces">Totally geodesic subspaces</h2> <p>Every totally geodesic submanifold of the complex hyperbolic space of dimension n is one of the following : </p> <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 n {\displaystyle n} </li></ul> <p>In particular, there is no codimension 1 totally geodesic subspace of the complex hyperbolic space. </p> <h2 id="link-with-other-metrics-on-the-ball">Link with other metrics on the ball</h2> <ul><li>On the unit ball, the complex hyperbolic metric coincides, up to some scalar renormalization, with the <a href="/facts/Bergman_metric/6Cuc9g6n">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="/facts/Kobayashi_metric/QEDNuxxu">Kobayashi metric</a>.</li></ul> <ul><li>Up to renormalization, the complex hyperbolic metric is <a href="/facts/K%25C3%25A4hler%25E2%2580%2593Einstein_metric/bHzK7qmU">Kähler-Einstein</a>, which means that its <a href="/facts/Ricci_curvature/qujou7QS">Ricci curvature</a> is a multiple of the metric.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Hyperbolic_space/kI7tnwdk">Hyperbolic space</a></li> <li>Quaternionic hyperbolic space</li></ul> <ul><li><a href="/facts/William_Goldman_(mathematician)/e5ChUlFb">Goldman, William M.</a> (1999). <i>Complex hyperbolic geometry</i>. Oxford: Clarendon Press. p. xx + 316. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853793-X.{{cite book}}: CS1 maint: publisher location (link)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Arthur Besse (1987), Einstein manifolds, Springer, p. 180. <a href="/wiki/Arthur_Besse" target="_blank">/wiki/Arthur_Besse</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Cano, Angel; Navarrete, Juan Pablo; Seade, José (2013). Complex Kleinian Groups. Progress in Mathematics. Vol. 303. doi:10.1007/978-3-0348-0481-3. ISBN 978-3-0348-0480-6. <a href="978-3-0348-0480-6" target="_blank">978-3-0348-0480-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Kobayashi, Shōshichi; Nomizu, Katsumi (1996). Foundations of differential geometry, vol. 2. New York: Wiley. ISBN 0-471-15733-3. OCLC 34259751. <a href="0-471-15733-3" target="_blank">0-471-15733-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kobayashi, Shōshichi; Nomizu, Katsumi (1996). Foundations of differential geometry, vol. 2. New York: Wiley. ISBN 0-471-15733-3. OCLC 34259751. <a href="0-471-15733-3" target="_blank">0-471-15733-3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>