Hyperbolic space

In geometry , the hyperbolic space is a space with constant negative curvature. It fulfills the axioms of Euclidean geometry with the exception of the axiom of parallels . The two-dimensional hyperbolic space with constant curvature is called the hyperbolic plane . ${\displaystyle -1}$

definition

Be a natural number. The n-dimensional hyperbolic space is the n-dimensional , simply connected , complete Riemann manifold with constant intersection curvature . ${\displaystyle n}$ ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle -1}$

The existence of the n-dimensional hyperbolic space results from the models given below, the uniqueness from Cartan's theorem.

Occasionally, the term hyperbolic space is also used more generally for hyperbolic spaces in the Gromov sense . In the following, however, this article only considers the hyperbolic space with section curvature −1. At the end of the article, other (partly incompatible) uses of the term "hyperbolic space" in mathematics are listed. ${\displaystyle \delta }$

Uniqueness

From a theorem of Elie Cartan it follows that the n-dimensional hyperbolic space is unique except for isometry. In particular, the models of n-dimensional hyperbolic space given below are all isometric to one another.

properties

Hyperbolic triangle

For every geodesic and every point there are infinitely many too disjoint geodesics through . ${\displaystyle L}$ ${\displaystyle P\not \in L}$ ${\displaystyle L}$ ${\displaystyle P}$

The interior angle sum of triangles is always less than . Is the area of ​​a triangle , where are interior angles. ${\displaystyle \pi }$ ${\displaystyle \pi -(\alpha +\beta +\gamma )}$ ${\displaystyle \alpha ,\beta ,\gamma }$

trigonometry

The formulas of hyperbolic trigonometry apply:

${\displaystyle {\frac {\sin \alpha }{\sinh a}}={\frac {\sin \beta }{\sinh b}}={\frac {\sin \gamma }{\sinh c}}}$

and

${\displaystyle \cosh c=\cosh a\cosh b-\sinh a\sinh b\cos \gamma ,\,}$

where are the interior angles of a triangle and the lengths of the opposite sides. ${\displaystyle \alpha ,\beta ,\gamma }$ ${\displaystyle a,b,c}$

Exponential growth

The volume of a ball by the radius is ${\displaystyle r}$

${\displaystyle {\frac {2\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}\right)}}\int _{0}^{r}\sinh ^{n-1}(\rho )\,d\rho }$,

it thus grows exponentially with the radius.

Isometrics

Geodetic half-lines in are called asymptotic if they have a finite distance. This defines an equivalence relation on the set of geodetic half-lines. The boundary at infinity is the set of equivalence classes of geodetic half-straight lines parameterized for arc length. Every isometry can be continued indefinitely on the edge . ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \partial _{\infty }\mathbb {H} ^{n}}$ ${\displaystyle f\colon \mathbb {H} ^{n}\rightarrow \mathbb {H} ^{n}}$ ${\displaystyle \partial _{\infty }H^{n}}$

The isometries of the hyperbolic space fall into the following classes (apart from the identity mapping, disjoint):

• elliptical :has a fixed point in, ${\displaystyle f}$ ${\displaystyle \mathbb {H} ^{n}}$
• loxodromic :has no fixed point in, but leaves two points inand the geodesic connecting them invariant, ${\displaystyle f}$ ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \partial _{\infty }\mathbb {H} ^{n}}$
• parabolic :leaves a pointand its horospheres invariant. ${\displaystyle f}$ ${\displaystyle p\in \partial _{\infty }\mathbb {H} ^{n}}$

The group of isometries is isomorphic to . ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle O^{+}(n,1)}$

Models

Poincaré half-space model

Division of the upper half plane into isometric geodetic heptagons

The half space

${\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}>0\right\}\subset \mathbb {R} ^{n}}$

with the Riemannian metric

${\displaystyle {\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{x_{n}^{2}}}}$

is a model of hyperbolic space.

For it is also known as the Poincaré half-plane model. ${\displaystyle n=2}$

Poincaré ball model

Division of the circular disk: areas of the same color are isometric to one another in the Poincaré ball model.

The open sphere

${\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}$

with the Riemannian metric

${\displaystyle 4{\frac {(dx_{1})^{2}+\ldots +(dx_{n})^{2}}{\left(1-x_{1}^{2}-\ldots -x_{n}^{2}\right)^{2}}}}$

is a model of hyperbolic space.

For it is also known as the Poincaré circular disk model. ${\displaystyle n=2}$

Hyperboloid model

Consider the one with the pseudo-Riemannian metric . ${\displaystyle \mathbb {R} ^{n+1}}$ ${\displaystyle -(dx_{1})^{2}+(dx_{2})^{2}+\ldots +(dx_{n+1})^{2}}$

The hyperboloid

${\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {R} ^{n+1}:-x_{1}^{2}+x_{2}^{2}\ldots +x_{n+1}^{2}=1,x_{1}>0\right\}\subset \mathbb {R} ^{n+1}}$

with the induced metric is a model of hyperbolic space.

Projective model

Division of the circular disc into triangles and heptagons, which in the Beltrami-Klein model are geodetic and each isometric to one another.

Let the canonical projection on the projective space be , then the projective model of the hyperbolic space is obtained as an image of the hyperboloid below . ${\displaystyle p\colon \mathbb {R} ^{n+1}\setminus \{0\}\rightarrow \mathbb {R} P^{n}}$ ${\displaystyle p}$

Once identified , the projective model corresponds to the set ${\displaystyle \mathbb {R} P^{n}=\mathbb {R} ^{n}\cup \mathbb {R} P^{n-1}}$

${\displaystyle \left\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{1}^{2}+\ldots +x_{n}^{2}<1\right\}\subset \mathbb {R} ^{n}}$.

Distances are calculated according to the Hilbert metric

${\displaystyle d(p,q)={\frac {1}{2}}\log {\frac {|qa||bp|}{|pa||bq|}}}$,

where the absolute bars should stand for Euclidean distances and the points of intersection of the geodesics through with the unit sphere. ${\displaystyle a,b}$ ${\displaystyle p,q}$

history

The projective model, the Poincaré ball model and the Poincaré half-space model were constructed by Eugenio Beltrami in 1868 , all three as images of another (so-called "hemispherical") model under suitable isometrics. The Poincaré ball model had already been investigated by Liouville for 1850 and the projective model appeared in a work by Cayley on projective geometry in 1859 , but without establishing a connection to hyperbolic geometry. ${\displaystyle n=2}$

Previously, Nikolai Ivanovich Lobatschewski and János Bolyai had developed a theory of hyperbolic space based on axioms and formally derived many of its properties. However, it was only with the models given by Beltrami that the proof was provided that the hyperbolic geometry is free of contradictions.

Henri Poincaré discovered that hyperbolic geometry occurs naturally in the study of differential equations and in number theory (in the study of square shapes ). In connection with the study of ternary square shapes, he first used the hyperboloid model in 1881.

Homogeneous space

The hyperbolic space is the homogeneous space

${\displaystyle \mathbb {H} ^{n}=O(n,1)/(O(n)\times O(1))=O_{0}(n,1)/O(n)=SO_{0}(n,1)/SO(n),}$

where the connected component denotes the one in . ${\displaystyle (S)O_{0}(n,1)}$ ${\displaystyle (S)O(n,1)}$

This means that hyperbolic geometry is a geometry in the sense of Felix Klein's Erlanger program .

For you also have the representations ${\displaystyle n=2,3}$

${\displaystyle \mathbb {H} ^{2}=SL(2,\mathbb {R} )/SO(2)}$

${\displaystyle \mathbb {H} ^{3}=SL(2,\mathbb {C} )/SU(2)}$.

Embedding in Euclidean space

The hyperbolic space has an isometric - embedding in the Euclidean space . ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle C^{\infty }}$ ${\displaystyle \mathbb {R} ^{4n-3}}$

Other uses of the term "hyperbolic space"

• In metric geometry , hyperbolic spaces in the Gromov sense (also referred to as Gromov hyperbolic spaces) are a class of metric spaces, to which, among other things, simply connected manifolds of negative sectional curvature (especially hyperbolic space) belong. Finitely generated groups are called hyperbolic groups if their Cayley graph is a hyperbolic space. ${\displaystyle \delta }$ ${\displaystyle \delta }$
• In the theory of symmetric spaces there are, in addition to the hyperbolic spaces considered in this article, which are often referred to as real-hyperbolic spaces in this context, the complex hyperbolic and quaternionic-hyperbolic spaces as well as the Cayley hyperbolic plane . These are defined for or as using the induced Riemannian metric. ${\displaystyle \mathbb {K} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} }$ ${\displaystyle \mathbb {O} }$ ${\displaystyle \left\{(x_{1},\ldots ,x_{n+1})\in \mathbb {K} ^{n+1}:x_{1}^{2}-x_{2}^{2}\ldots -x_{n+1}^{2}=1\right\}\subset \mathbb {K} ^{n+1}}$
• In incidence geometry , a hyperbolic space is an arranged incidence space with a congruence relation and the property that every level with the induced arrangement and congruence relation is a hyperbolic level in the sense of Karzel-Sörensen-Windelberg. In particular, there is the concept of finite hyperbolic spaces in finite geometry.
• In complex analysis , a complex manifold is called Brody hyperbolic if every holomorphic map is constant. This applies in particular to the complex structure given by the Poincaré circular disk model on the hyperbolic level, see Liouville's theorem . ${\displaystyle X}$ ${\displaystyle f\colon \mathbb {C} \rightarrow X}$
• Also in complex analysis, a complex manifold is called Kobayashi hyperbolic (or just hyperbolic) if the Kobayashi pseudo-metric is a metric. For compact complex manifolds, Brody hyperbolicity and Kobayashi hyperbolicity are equivalent.
• In complex differential geometry, Kähler manifolds are called Kähler hyperbolic if the raised Kähler form of the universal superposition is the differential of a restricted differential form. ${\displaystyle (M,\omega )}$ ${\displaystyle {\widetilde {\omega }}}$ ${\displaystyle {\widetilde {M}}}$
• In homotopy theory , a hyperbolic space is a topological space with . Here indicates the i-th homotopy group and its rank . This definition is unrelated to the one discussed in this article. ${\displaystyle X}$ ${\displaystyle \sum _{i}rk(\pi _{i}X)=\infty }$ ${\displaystyle \pi _{i}X}$ ${\displaystyle rk}$

literature

• Eugenio Beltrami: Saggio di interpretazione della geometria non-euclidea. Giornale Matemat. 6: 284-312 (1868)
• Eugenio Beltrami: Teoria fondamentale degli spazii di curvatura constante. Ann. Mat. Ser. II 2 (1868-69), 232-255, doi: 10.1007 / BF02419615 .
• Felix Klein: About the so-called non-Euclidean geometry Math. Ann. 4 (1871), 573-625, doi: 10.1007 / BF01443189 .
• Henri Poincaré: Théorie des groupes fuchsiens. Acta Math. 1 (1882), 1-62 pdf
• Henri Poincaré: Mémoire sur les groupes kleinéens. Acta Math. 3 (1883), 49-92 pdf
• Henri Poincaré: Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques. Assoc. Franc. Compt. Rend. 1881, 132-138 pdf

The 6 above works are translated into English into:

• Stillwell, John: Sources of hyperbolic geometry. History of Mathematics, 10. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1996. x + 153 pp. ISBN 0-8218-0529-0

Web links

• Cannon, Floyd, Kenyon, Parry: Hyperbolic Geometry (PDF; 425 kB)

Individual evidence

1. ↑ Oláh-Gál: The n-dimensional hyperbolic space in E ^{4n − 3} . Publ. Math. Debrecen 46 (1995), no. 3-4, 205-213.
2. ↑ Karzel-Sörensen-Windelberg: Introduction to Geometry. Göttingen 1973