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The Poincaré model of hyperbolic geometry in an arbitrary real inner product space and an elementary construction of hyperbolic triangles with prescribed angles. (English) Zbl 1204.51021

In a hyperbolic triangle of the hyperbolic plane the sum of angles is less than \(\pi\) and for given values \(\alpha,\beta,\gamma> 0\) with \(\alpha+\beta+\gamma<\pi\) there is always a hyperbolic triangle with these angles. The authors replace in their highly interesting paper the hyperbolic plane by the hyperbolic space over an arbitrary real inner product space \((X,\cdot)\) of finite or infinite dimension \(\geq 2\), thus working with structures playing the basic role in the dimension-free written book “Classical geometries in modern contexts. Geometry of real inner product spaces.” [2nd ed. (Basel: Birkhäuser) (2008; Zbl 1135.51002)] of the reviewer. The authors base their theory on differentiable curves and their tangents of spaces \((X,\cdot)\) and on conformal (angle preserving) mappings. They prove that Möbius transformations on spaces \((X,\cdot)\) are conformal (angle preserving) and that hyperbolic and Euclidean angles in Poincare’s model coincide, as well-known for the elementary situation. Poincaré’s model consists here of the set \(D(r)\) of points \(x\in X\) with \(x\cdot x< r^2\) for a suitable chosen \(r> 1\), and of the hyperbolic lines \(C\cap D(r)\), where \(C\) are Möbius circles perpendicular to the ball \(\{x\in x\mid x\cdot x= r^2\}\). Now showing the usefulness of the dimension-free working and concepts, hyperbolic triangles in the Poincaré model are obtained by elementary geometric constructions with arbitrarily prescribed angles.

MSC:

51M09 Elementary problems in hyperbolic and elliptic geometries

Citations:

Zbl 1135.51002
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