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Hyperbolic geometry, Coxeter groups, polylogarithm identities Ruth Kellerhals (born 17 July 1957) is a Swiss mathematician at the University of Fribourg , whose field of study is hyperbolic geometry , geometric group theory and polylogarithm identities.
Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...
The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature) is -hyperbolic. If we scale the Riemannian metric by a factor λ > 0 {\displaystyle \lambda >0} then the distances are multiplied by λ {\displaystyle \lambda } and thus we get a space that is λ ⋅ δ {\displaystyle \lambda \cdot \delta } -hyperbolic.
In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...
While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic -manifold (for >) is a point, for a hyperbolic surface of genus > there is a moduli space of dimension that parameterizes all metrics of constant curvature (up to diffeomorphism), a fact essential for Teichmüller theory.
Two-dimensional hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Most hyperbolic surfaces have a non-trivial fundamental group π 1 = Γ; the groups that arise this way are known as Fuchsian groups.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
In geometry, the order-5 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,5}, constructed from five pentagons around every vertex. As such, it is self-dual .