Hyperbolic distances in Hilbert spaces

Benz, Walter

doi:10.1007/s000100050003

Cited by 6 publications

(17 citation statements)

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“…With these notations, we consider the set H = H ∪ ∞ equipped with the following topology: U ⊂ H is an open set if and only if U ∩ H is an open set and H\U is a bounded set in H, if ∞ ∈ U . In this section we will recall the classical properties of the Möbius transformations of H which the reader can find in ( [3], [2] and [11] ). We first introduce the following notations:…”

Section: Möbius Transformationsmentioning

confidence: 99%

Möbius transformations and the configuration space of a Hilbert snake

Pelletier

Saffidine

Bensalem

2015

Bulletin des Sciences Mathématiques

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Abstract. The purpose of this paper is to give a simpler proof to the problem of controllability of a Hilbert snake [13]. Using the action of the Möbius group of the unite sphere on the configuration space, in the context of a separable Hilbert space. We give a generalization of the Theorem of accessibility contained in [9] and [14] for articulated arms and snakes in a finite dimensional Hilbert space.classification: 22F50, 34C40, 34H, 53C17, 53B30, 53C50, 58B25, 93B03.

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Section: Möbius Transformationsmentioning

confidence: 99%

Möbius transformations and the configuration space of a Hilbert snake

Pelletier

Saffidine

Bensalem

2015

Bulletin des Sciences Mathématiques

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“…We then know (compare [4], p. 18) that orthogonal mappings are injective and linear and that (1) holds for all x, y ∈ X. Denote by O (X) the set of all orthogonal mappings of X into itself, and by O (X) the group of all surjective orthogonal mappings of X. If ω ∈ O (X) then…”

Section: Möbius Balls Möbius Transformations Inversionsmentioning

confidence: 99%

“…Kowalsky [16], p. 181), hence avoiding transfinite methods, which could be considered as somewhat strange in the context of geometries of Klein's Erlangen programme. By the way, in the same spirit as in the present paper, we dealt with other geometries of this programme, like Lie Sphere Geometry [3] and Hyperbolic Geometry [4].…”

Section: Introductionmentioning

confidence: 99%

M�bius sphere geometry in inner product spaces

Benz

2003

Aequationes Mathematicae

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We develop Möbius sphere geometry for arbitrary euclidean spaces (i.e. real inner product spaces or real pre-Hilbert spaces) X of (finite or infinite) dimension at least 2. All Möbius transformations of X are determined, especially those which are involutorial. Moreover, M -transformations are characterized within the group of Lie transformations of X. We prove that the 4-point-invariants must be functions of the cross ratio. Stereographic projection from a hypersphere of X ⊕ R onto X ∪ {∞} is introduced, and also Poincaré's model of hyperbolic geometry with respect to an M -ball B and one of the sides Σ of B. All bijections of Σ preserving hyperbolic distances are determined: they are exactly the Möbius transformations µ satisfying µ (Σ) = Σ. An isomorphism between the models of Poincaré and Weierstrass of hyperbolic geometry over X is established. Mathematics Subject Classification (2000). 51B10, 51M10, 51N25, 39B52.

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“…For our purposes now, we will generalize our Theorem 2 in [5]. Denote by R ≥0 the set of all real numbers r ≥ 0 and by R >0 the set R ≥0 \{0}.…”

Section: Distance Functions Of X 0 Xmentioning

confidence: 99%

“…Among the results of this note are a characterization of the function ε (x, y), more precisely a functional equations approach to ε (x, y), and, moreover, a similar approach to σ (x, y) (see Theorems 3,4,5,6). We furthermore deal with elliptic and spherical geometry on the basis of the distance notions (2), (3), respectively: points are defined, lines and hyperspheres, and the isometries of the geometries in question are determined.…”

Section: Introductionmentioning

confidence: 99%