An axiomatic analysis of the Droz-Farny Line Theorem

We analyze an elementary theorem of Euclidean geometry, the Droz-Farny Line Theorem, from the point of view of the foundations of geometry. We start with an elementary synthetic proof which is based on simple properties of the group of motions. The proof reveals that the Droz-Farny Line Theorem is a special case of the Theorem of Goormatigh which is, in turn, a special case of the Counterpairing Theorem of Hessenberg. An axiomatic analysis in the sense of Hilbert [14] and Bachmann [2] leads to a study of different versions of the theorems (e.g., of a dual version or of an absolute version, which is valid in absolute geometry) and to a new axiom system for the associated very general plane absolute geometry (the geometry of pencils and lines). In the last section the role of the theorems in the foundations of geometry is discussed.     

The Funk and Hilbert geometries for spaces of constant curvature

The goal of this paper is to introduce and to study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets of the hyperbolic space $\mathbb H ^n$ H n and of the sphere $S^n$ S n . We highlight some striking similarities among the three cases (Euclidean, spherical and hyperbolic) which hold at least at a formal level. The proofs of the basic properties of the classical Funk metric on subsets of $\mathbb R ^n$ R n use similarity properties of Euclidean triangles which of course do not hold in the non-Euclidean cases. Transforming the side lengths of triangles using hyperbolic and circular functions and using some non-Euclidean trigonometric formulae, the Euclidean similarity techniques are transported into the non-Euclidean worlds. We start by giving three representations of the Funk metric in each of the non-Euclidean cases, which parallel known representations for the Euclidean case. The non-Euclidean Funk metrics are shown to be Finslerian, and the associated Finsler norms are described. We then study their geodesics. The Hilbert geometry of convex sets in the non-Euclidean constant curvature spaces $S^n$ S n and $\mathbb H ^n$ H n is then developed by using the properties of the Funk metric and by introducing a non-Euclidean cross ratio. In the case of Euclidean (respectively spherical, hyperbolic) geometry, the Euclidean (respectively spherical, hyperbolic) geodesics are Funk and Hilbert geodesics. This leads to a formulation and a discussion of Hilbert’s Problem IV in the non-Euclidean settings. Projection maps between the spaces $\mathbb R ^n, \mathbb H ^n$ R n , H n and the upper hemisphere establish equivalences between the Hilbert geometries of convex sets in the three spaces of constant curvature, but such an equivalence does not hold for Funk geometries.     

On Algebraic Structures Related to Beltrami–Klein Model of Hyperbolic Geometry

A new approach to the algebraic structures related to hyperbolic geometry comes from Einstein’s special theory of relativity in 1988 (cf. Ungar, in Found Phys Lett 1:57–89, 1988). Ungar employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry (cf. Ungar, in Math Appl 49:187–221, 2005). Another approach is from Karzel for algebraization of absolute planes in the sense of Karzel et al. (Einführung in die Geometrie, 1973). In this paper we are going to develop a formulary for the Beltrami–Klein model of hyperbolic plane inside the unit circle ${\mathbb D}$ of the complex numbers ${\mathbb C}$ with geometric approach of Karzel.     

On Trigonometry in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Math. Appl. 49:187–221, 2005) employed the binary operation of Einsteins velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry in full analogy with Euclidean geometry. We use the isomorphism between ${(\mathbb R,+,\cdot)}$ and ${((-1,1),\oplus,\otimes)}$ in Beltrami–Klein model of hyperbolic geometry for similar results.     

Two Kneser–Poulsen-type inequalities in planes of constant curvature

We show that the perimeter of the convex hull of finitely many disks lying in the hyperbolic or Euclidean plane, or in a hemisphere does not increase when the disks are rearranged so that the distances between their centers do not increase. This generalizes the theorem on the monotonicity of the perimeter of the convex hull of a finite set under contractions, proved in the Euclidean plane by V. N. Sudakov [8], R. Alexander [1], V. Capoyleas and J. Pach [3]. We also prove that the area of the intersection of finitely many disks in the hyperbolic plane does not decrease after such a contractive rearrangement. The Euclidean analogue of the latter statement was proved by K. Bezdek and R. Connelly [2]. Both theorems are proved by a suitable adaptation of a recently published method of I. Gorbovickis [4].     

A Clifford Algebraic Approach to Line Geometry

In this paper we combine methods from projective geometry, Klein’s model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use is constructed as homogeneous model for the five-dimensional real projective space \({\mathbb {P}^5 (\mathbb{R})}\) where Klein’s quadric \({M^4_2}\) defines the quadratic form. We discuss all entities that can be represented naturally in this homogeneous Clifford algebra model. Projective automorphisms of Klein’s quadric induce projective transformations of \({\mathbb {P}^3 (\mathbb{R})}\) and vice versa. Cayley-Klein geometries can be represented by Clifford algebras, where the group of Cayley-Klein isometries is given by the Pin group of the corresponding Clifford algebra. Therefore, we examine the versor group and study the correspondence between versors and regular projective transformations represented as 4 × 4 matrices. Furthermore, we give methods to compute a versor corresponding to a given projective transformation.     

Geometric algebra of strictly convex Minkowski planes

Imposing geometric or group-theoretical conditions on left reflections or the group \({\mathfrak{G}}\) generated by them, we obtain many characterizations of the Euclidean plane and of Radon planes within the framework of strictly convex Minkowski planes. In particular, Bachmann’s view of geometry provides a rich source of pertinent conditions on \({\mathfrak{G}}\) . A special role in characterizing the Euclidean plane and Radon planes is played by the shape of the locus of images of a point x under the set of left reflections in lines having a point distinct from x in common.     

Defect and Area in Beltrami–Klein Model of Hyperbolic Geometry

Ungar (Beyond the Einstein addition law and its gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrouector Spaces, 2001; Comput Math Appl 49:187–221, 2005; Comput Math Appl 53, 2007) introduced into hyperbolic geometry the concept of defect based on relativity addition of A. Einstein. Another approach is from Karzel (Resultate Math. 47:305–326, 2005) for the relation between the K-loop and the defect of an absolute plane in the sense (Karzel in Einführung in die Geometrie, 1973). Our main concern is to introduce a systematical exact definition for defect and area in the Beltrami–Klein model of hyperbolic geometry. Combining the ideas and methods of Karzel and Ungar give an elegant concept for defect and area in this model. In particular we give a rigorous and elementary proof for the defect formula stated (Ungar in Comput Math Appl 53, 2007). Furthermore, we give a formulary for area of circle in the Beltrami–Klein model of hyperbolic geometry.     

Hypercomplex limits of pluricomplex structures and the Euclidean limit of hyperbolic monopoles

We discuss the Euclidean limit of hyperbolic $SU(2)$ -monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.     

Inversion with respect to a hypercycle of a hyperbolic plane of positive curvature

Inversion with respect to a hypercycle of a hyperbolic plane \({\widehat{H}}\) of positive curvature is investigated. The plane \({\widehat{H}}\) is the projective Cayley–Klein model of two-dimensional de Sitter’s space. One of four analogs of a Euclidean circle on the plane \({\widehat{H}}\) is a hypercycle. The formulae of inversion with respect to the hypercycle in a canonical frame of the first type are derived. The main properties of this inversion are proved.     

Unit Balls, Lorentz Boosts, and Hyperbolic Geometry

The bounded symmetric spaces naturally associated with the Poincaré and Beltrami-Klein models of hyperbolic geometry on the open unit ball B in ${\mathbb{R}^n}$ and with the automorphism group of biholomorphic maps on the open ball in ${\mathbb{C}^n}$ give rise by a standard construction to specialized loop structures (nonassociative groups), which we use to define canonical metrics, called rapidity metrics. We show that this rapidity metric agrees with the classical Poincaré metric resp. the Cayley-Klein metric resp. the Bergman metric. We introduce the Lorentz boost of vectors in B, which turns out to be a loop isomorphism. It induces a similarity of metrics between the rapidity metric of the Einstein or Möbius loop and the trace metric on positive definite matrices restricted to the Lorentz boosts.     

Circle Geometries Modeled in Projective Lines over {{\mathbb{R}}^2} -rings

In this paper, we consider classical circle geometries and connect them with places of planar Cayley–Klein geometries. There are, in principle, only three types of $ {{\mathbb{R}}^2} $ -ring structures and, thus, only three types of corresponding circle geometries. Thus, each generalization to non-Euclidean planes turns out to be just another representation of the classical Euclidean cases. We believe that even the Euclidean cases of circle geometries comprise, in principle, already all non-Euclidean cases. Representations of such non-Euclidean circle geometries might also be of interest in themselves. For example, among the planar Cayley–Klein geometries, the quasi-elliptic and quasi-hyperbolic geometry usually are neglected. They can be treated similarly to the isotropic Möbius geometry by suitable projections of the Blaschke cylinder.     

Harmonic Analysis on the Möbius Gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball \({{\mathbb {B}}_{t}^{n}}\), a model of the \(n\)-dimensional real hyperbolic space. The Poincaré ball \({{\mathbb {B}}_{t}^{n}}\) is the open ball of the Euclidean \(n\)-space \(\mathbb {R}^n\) with radius \(t >0\), centered at the origin of \(\mathbb {R}^n\) and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in \(\mathbb {R}^n.\) For any \(t>0\) and an arbitrary parameter \(\sigma \in \mathbb {R}\) we study the \((\sigma ,t)\)-translation, the \((\sigma ,t)\)-convolution, the eigenfunctions of the \((\sigma ,t)\)-Laplace–Beltrami operator, the \((\sigma ,t)\)-Helgason Fourier transform, its inverse transform and the associated Plancherel’s Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when \(t \rightarrow +\infty \) the resulting hyperbolic harmonic analysis on \({{\mathbb {B}}_{t}^{n}}\) tends to the standard Euclidean harmonic analysis on \(\mathbb {R}^n,\) thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on \({{\mathbb {B}}_{t}^{n}}\).     

Normal subgroups in the Cremona group

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $ \mathbb{P}_{\mathbf{k}}^2 $ is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.     

Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries

We consider two Riemannian geometries for the manifold \({\mathcal{M }(p,m\times n)}\) of all \(m\times n\) matrices of rank \(p\) . The geometries are induced on \({\mathcal{M }(p,m\times n)}\) by viewing it as the base manifold of the submersion \(\pi :(M,N)\mapsto MN^\mathrm{T}\) , selecting an adequate Riemannian metric on the total space, and turning \(\pi \) into a Riemannian submersion. The theory of Riemannian submersions, an important tool in Riemannian geometry, makes it possible to obtain expressions for fundamental geometric objects on \({\mathcal{M }(p,m\times n)}\) and to formulate the Riemannian Newton methods on \({\mathcal{M }(p,m\times n)}\) induced by these two geometries. The Riemannian Newton methods admit a stronger and more streamlined convergence analysis than the Euclidean counterpart, and the computational overhead due to the Riemannian geometric machinery is shown to be mild. Potential applications include low-rank matrix completion and other low-rank matrix approximation problems.     

Reverse Bonnesen style inequalities in a surface \mathbb{X}_\varepsilon ^2 of constant curvature

We investigate the isoperimetric deficit upper bound, that is, the reverse Bonnesen style inequality for the convex domain in a surface $\mathbb{X}_\varepsilon ^2$ of constant curvature ? via the containment measure of a convex domain to contain another convex domain in integral geometry. We obtain some reverse Bonnesen style inequalities that extend the known Bottema’s result in the Euclidean plane $\mathbb{E}^2$ .     

Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53?C70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561?C598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric $\mathbb{M}^{d}$ and the hyperbolic metric ? ^d . This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among $\mathbb{E}^{d}$ , cones in $\mathbb{E}^{d+1}$ , $\mathbb{S}^{d}$ , $\mathbb{M}^{d}$ , and ? ^d . We also consider the further extensions associated with the other Cayley?CKlein geometries overlaid on the shared underlying projective geometry.     

Proper Harmonic Maps from Hyperbolic Riemann Surfaces into the Euclidean Plane

Let Σ be a compact Riemann surface and D ₁ , . . . , D _n a finite number of pairwise disjoint closed disks of Σ. We prove the existence of a proper harmonic map into the Euclidean plane from a hyperbolic domain Ω containing ${\Sigma \backslash \cup_{j=1}^n D_j}$ and of its topological type. Here, Ω can be chosen as close as necessary to ${\Sigma \backslash \cup_{j=1}^n D_j}$ .