Hyperbolic level

The hyperbolic plane is an object from the mathematical subfield of geometry , more precisely from hyperbolic geometry . In addition to the Euclidean plane and the sphere, this geometrical object belongs to the model spaces of area theory . Because it has the constant Gaussian or sectional curvature . Euclidean space has curvature and the sphere has curvature . In contrast to these two spaces, the hyperbolic plane as a whole cannot be embedded in Euclidean space . ${\ displaystyle -1}$ ${\ displaystyle 0}$ ${\ displaystyle 1}$ ${\ displaystyle \ mathbb {R} ^ {3}}$

definition

The hyperbolic plane is defined as the 2-dimensional hyperbolic space , i.e. as a two-dimensional , simply connected , complete Riemannian manifold with constant sectional curvature . ${\ displaystyle \ mathbb {H} ^ {2}}$ ${\ displaystyle -1}$

The hyperbolic plane can be characterized with Poincaré's half-space model. If one equips the half-plane with the metric , one obtains the hyperbolic plane. ${\ displaystyle \ {(x, y) \ in \ mathbb {R} ^ {2}: y> 0 \}}$ ${\ displaystyle g = y ^ {- 2} (\ mathrm {d} x ^ {2} + \ mathrm {d} y ^ {2})}$

In terms of the Erlangen program , the hyperbolic plane can be interpreted as the geometry of the couple . ${\ displaystyle (SL (2, \ mathbb {R}), SO (2))}$

The hyperbolic plane can be characterized axiomatically by the fact that it fulfills all axioms of Euclidean geometry with the exception of the axiom of parallels and also the axiom that a line g and a point P (which does not lie on g) at least two lines (h and i) which go through P and are parallel (ie disjoint) to g.

Other uses of the term "hyperbolic plane"

In incidence geometry , a (finite) hyperbolic plane is a (finite) set of "points" H with certain subsets as "straight lines", which satisfy the following axioms:

1. two different points each belong to exactly one straight line,

2. if a point P does not belong to a line l, then there are at least two lines containing P disjoint to l,

3. if a set of points S contains three points that are not on a straight line and contains all points on a straight line through two points each in S, then S = H.

In the theory of symmetrical spaces there is, in addition to the hyperbolic plane (referred to in this context as the real hyperbolic plane), the complex hyperbolic, quaternionic hyperbolic and Cayley hyperbolic plane.

In the work of Helmut Karzel and his students, "hyperbolic level" denotes an arranged incidence space with a congruence relation that fulfills certain axioms. This term axiomatizes the arrangement and incidence properties of the hyperbolic level defined above without referring to its metrics.

The 2-dimensional square space with is called the hyperbolic plane . This definition is not directly related to that defined above. ${\ displaystyle (V, q)}$ ${\ displaystyle q (x, y) = xy}$

Web links

Commons : Models of the hyperbolic plane - collection of images, videos and audio files

Individual evidence

↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 7.
↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , pp. 7 , 38.

[Lee7-1] John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 7.

[Lee7u38-2] John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , pp. 7 , 38.