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Walter Benz - One of the best experts on this subject based on the ideXlab platform.
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Sphere Geometries of Möbius and Lie
Classical Geometries in Modern Contexts, 2012Co-Authors: Walter BenzAbstract:Also in this chapter X denotes a Real Inner Product space of arbitrary (finite or infinite) dimension ≥ 2.
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Euclidean and Hyperbolic Geometry
Classical Geometries in Modern Contexts, 2012Co-Authors: Walter BenzAbstract:X designates again an arbitrary Real Inner Product space containing two linearly independent elements. As throughout the whole book, we do not exclude the case that there exists an infinite and linearly independent subset of X.
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δ–Projective Mappings, Isomorphism Theorems
Classical Geometries in Modern Contexts, 2012Co-Authors: Walter BenzAbstract:Let (X, δ) and (V, e) be arbitrary Real Inner Product spaces each containing at least two linearly independent elements. However, as in the earlier chapters we do not exclude the case that there exist infinite linearly independent subsets of X or V.
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A Representation of Hyperbolic Motions Including the Infinite-Dimensional Case
Results in Mathematics, 2011Co-Authors: Walter BenzAbstract:Let X be a Real Inner Product space of (finite or infinite) dimension ≥ 2, O ( X ) be its group of all surjective (hence bijective) orthogonal transformations of X , T ( X ) be the set of all hyperbolic translations of X and M ( X , h yp ) be the group of all hyperbolic motions of X . The following theorem will be proved in this note. Every $${\mu\in M(X,{\mbox hyp})}$$ has a representation μ = T · ω with uniquely determined $${T\in T(X)}$$ and uniquely determined $${\omega\in O(X)}$$ .
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A characterization of dimension-free hyperbolic geometry and the functional equation of 2-point invariants
Aequationes mathematicae, 2010Co-Authors: Walter BenzAbstract:Let X be a Real Inner Product space of arbitrary (finite or infinite) dimension ≥ 2. Define $$P:=\{x\in X\mid\lVert x\rVert < 1\}\qquad\qquad(1)$$ and G to be the group of all bijections of P such that the images and pre-images of the following sets, called P -lines, $$(a,b):=\left\{x\in X\backslash\{a,b\} \mid\lVert a-b\rVert=\lVert a-x\rVert+\lVert x-b\rVert\right\},\qquad\qquad(2)$$ $${a,b\in X,\,a\neq b,\| a\|=1=\|b\|}$$ , are again P -lines. Observe that ( a , b ) is given by the segment $$(a,b)=\{a +\varrho(b-a)\mid 0 < \varrho < 1\},$$ where the two points a ≠ b are on the ball B (0, 1). In Theorem 1 we prove that the geometry ( P , G ) is isomorphic to the hyperbolic geometry ( X , M ( X , hyp)) over X (see Sect. 1). In Theorem 2 we solve the Functional Equation of 2-point invariants for ( P , G ), showing that the notion of hyperbolic distance for ( P , G ) stemming from the isomorphism of Theorem 1 must be a basis of all its 2-point invariants. For definitions see the book [Benz in Classical Geometries in Modern Contexts. Geometry of Real Inner Product Spaces. Birkhäuser, Basel, 1st edn ( 2005 ) (2nd edn, 2007)], especially Sect. 2.12 and 5.11.
Karol Baron - One of the best experts on this subject based on the ideXlab platform.
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On Orthogonally Additive Functions With Big Graphs
Annales Mathematicae Silesianae, 2017Co-Authors: Karol BaronAbstract:Abstract Let E be a separable Real Inner Product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.
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On the set of orthogonally additive functions with orthogonally additive second iterate
Aequationes mathematicae, 2017Co-Authors: Karol BaronAbstract:Let E be a Real Inner Product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all orthogonally additive functions from E into E with the Tychonoff topology.
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On orthogonally additive injections and surjections
Commentationes Mathematicae, 2016Co-Authors: Karol BaronAbstract:Let \(E\) be a Real Inner Product space of dimension at least 2 and \(V\) a linear topological Hausdorff space. If \(\operatorname{card}E\leq \operatorname{card} V\), then the set of all orthogonally additive injections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology. If \(\operatorname{card}V\leq \operatorname{card}E\), then the set of all orthogonally additive surjections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology.
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On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions
Annales Mathematicae Silesianae, 2015Co-Authors: Karol BaronAbstract:We show that the solution to the orthogonal additivity problem in Real Inner Product spaces depends continuously on the given function and provide an application of this fact.
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Orthogonally additive bijections are additive
Aequationes mathematicae, 2015Co-Authors: Karol BaronAbstract:Any orthogonally additive injection of a Real Inner Product space of dimension at least 2 onto an abelian group is additive.
Abraham A. Ungar - One of the best experts on this subject based on the ideXlab platform.
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On the unification of hyperbolic and euclidean geometry
Complex Variables Theory and Application: An International Journal, 2004Co-Authors: Abraham A. UngarAbstract:The polar decomposition of Mobius transformation of the complex open unit disc gives rise to the Mobius addition in the disc and, more generally, in the ball. Mobius addition and Einstein addition in the ball of a Real Inner Product space are isomorphic gyrogroup operations that play in the hyperbolic geometry of the ball a role analogous to the role that ordinary vector addition plays in the Euclidean geometry of . Mobius (Einstein) addition governs the Poincare (Beltrami) ball model of hyperbolic geometry just as vector addition governs the standard model of Euclidean geometry. Accordingly, we show in this article that resulting analogies enable Euclidean and hyperbolic geometry to be unified
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The hyperbolic derivative in the Poincaré ball model of hyperbolic geometry
Journal of Mathematical Analysis and Applications, 2001Co-Authors: Graciela S. Birman, Abraham A. UngarAbstract:The generic Möbius transformation of the complex open unit disc induces a binary operation in the disc, called the Möbius addition. Following its introduction, the extension of the Möbius addition to the ball of any Real Inner Product space and the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincaré ball model of hyperbolic geometry. The Möbius gyrovector spaces that emerge provide the setting for the Poincaré ball model of hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Möbius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry. © 2001 Academic Press.
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The Einstein Gyrovector Space
Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, 2001Co-Authors: Abraham A. UngarAbstract:In this chapter we introduce scalar multiplication ⊗E in the Einstein gyrogroup (Vc,⊕E), obtaining the Einstein gyrovector space (Vc, ⊕E, ⊗E). This, in turn, results in the emergence of the hyperbolic analytic geometry of the Einstein gyrovector space, which turns out to be the familiar Beltrami ball model of hyperbolic geometry. The ball Vc is equipped with the coordinates it inherits from its Real Inner Product space V, relative to which gyrovectors are represented. We close the chapter with the observation that the unique hyperbolic ‘straight line’ called a geodesic, passing through two given points a, b ∈ Vc is the set of all points $$a{{\oplus }_{E}}\left( {{\ominus }_{E}}a{{\oplus }_{E}}b \right){{\otimes }_{E}}t $$ of Vc, t ∈ ℝ, ⊖Ea = −a, which is analogous to its counterpart in Euclidean analytic geometry.
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Extension of the Unit Disk Gyrogroup into the Unit Ball of Any Real Inner Product Space
Journal of Mathematical Analysis and Applications, 1996Co-Authors: Abraham A. UngarAbstract:Abstract The group of all holomorphic automorphisms of the complex unit disk consists of Mobius transformations involving translation-like holomorphic automorphisms and rotations. The former are called gyrotranslations . As opposed to translations of the complex plane, which are associative-commutative operations (i.e., their composition law is associative and commutative) forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations (i.e., their composition law is gyroassociative and gyrocommutative) forming a gyrogroup . The complex unit disk gyrogroup has previously been studied by the author ( Aequationes Math. 47 , 1994, 240–254). Employing analogies shared by complex numbers and linear transformations of vector spaces, we extend in this article the complex disk gyrogroup and its Mobius transformations into the ball of any Real Inner Product space and its generalized Mobius transformations. A gyrogroup is a mathematical object which first arose in the study of relativistic velocities which, under velocity addition, form a nongroup gyrogroup, as opposed to prerelativistic velocities, which form a group under velocity addition. It has been discovered that the mathematical regularity, seemingly lost in the transition from prerelativistic to relativistic velocities, is concealed in a relativistic effect known as Thomas precession. In its abstract context, Thomas precession is called Thomas gyration, giving rise to our “gyroterminology.” Our gyroterminology, developed by the author ( Amer. J. Phys. 59 , 1991, 824–834), involves terms like gyrogroups, gyroassociative-gyrocommutative laws, and gyroautomorphisms, in which we extensively use the prefix “gyro.”
Raymond Freese - One of the best experts on this subject based on the ideXlab platform.
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Intrinsic four-point properties
Journal of Geometry, 2014Co-Authors: Edward Andalafte, Raymond Freese, Brody Dylan Johnson, Rebecca LelkoAbstract:Many characterizations of euclidean spaces (Real Inner Product spaces) among metric spaces have been based on euclidean four point embeddability properties. Related “intrinsic” four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known four point embedding properties.
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Altitude properties and characterizations of Inner Product spaces
Journal of Geometry, 2000Co-Authors: Edward Andalafte, Raymond FreeseAbstract:New characterizations of Real Inner Product spaces (euclidean spaces) among metric spaces are obtained from familiar formulas expressing the altitude (height) of a triangle as a function of the lengths of its sides. Other properties related to the altitude of a triangle are also shown to result in characterizations of euclidean spaces, or euclidean and hyperbolic spaces.
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Pythagorean euclidean four point properties
Journal of Geometry, 1997Co-Authors: Edward Andalafte, Charles Diminnie, Raymond FreeseAbstract:Characterizations of Real Inner Product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize Real Inner Product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.
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Weak homogeneity of metric pythagorean orthogonality
Journal of Geometry, 1996Co-Authors: Edward Andalafte, Raymond FreeseAbstract:It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of Real Inner Product spaces among complete, convex, externally convex metric spaces is obtained.
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Weak additivity of metric pythagorean orthogonality
Journal of Geometry, 1995Co-Authors: Raymond Freese, Edward AndalafteAbstract:It is well known that the property of additivity of pythagorean orthogonality characterizes Real Inner Product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes Real Inner Product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.
Edward Andalafte - One of the best experts on this subject based on the ideXlab platform.
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Intrinsic four-point properties
Journal of Geometry, 2014Co-Authors: Edward Andalafte, Raymond Freese, Brody Dylan Johnson, Rebecca LelkoAbstract:Many characterizations of euclidean spaces (Real Inner Product spaces) among metric spaces have been based on euclidean four point embeddability properties. Related “intrinsic” four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known four point embedding properties.
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Altitude properties and characterizations of Inner Product spaces
Journal of Geometry, 2000Co-Authors: Edward Andalafte, Raymond FreeseAbstract:New characterizations of Real Inner Product spaces (euclidean spaces) among metric spaces are obtained from familiar formulas expressing the altitude (height) of a triangle as a function of the lengths of its sides. Other properties related to the altitude of a triangle are also shown to result in characterizations of euclidean spaces, or euclidean and hyperbolic spaces.
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Pythagorean euclidean four point properties
Journal of Geometry, 1997Co-Authors: Edward Andalafte, Charles Diminnie, Raymond FreeseAbstract:Characterizations of Real Inner Product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize Real Inner Product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.
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Weak homogeneity of metric pythagorean orthogonality
Journal of Geometry, 1996Co-Authors: Edward Andalafte, Raymond FreeseAbstract:It is known that euclidean or hyperbolic spaces are characterized among certain metric spaces by the property of linearity of the equidistant locus of pairs of points. In this paper, this linearity requirement is replaced by the requirement of convexity of the set of points which are metrically pythagorean orthogonal to a given segment at a given point. As a result a new characterization of Real Inner Product spaces among complete, convex, externally convex metric spaces is obtained.
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Weak additivity of metric pythagorean orthogonality
Journal of Geometry, 1995Co-Authors: Raymond Freese, Edward AndalafteAbstract:It is well known that the property of additivity of pythagorean orthogonality characterizes Real Inner Product spaces among normed linear spaces. In the present paper, a natural concept of additivity is introduced in metric spaces, and it is shown that a weakened version of this additivity of metric pythagorean orthogonality characterizes Real Inner Product spaces among complete, convex, externally convex metric spaces, providing a generalization of the earlier characterization.
