The invariants of a pair of directions in geometry [En1] and their interpretation

• Autorzy: Misiak A. , Stasiak E. , Szmuksta-Zawadzka M.
• Języki publikacji: EN

Abstrakty

EN

Solving a certain functional equation, we find all invariants of a pair of directions in n-dimensional pseudo-Euclidean geometry of index one [En1]. In (n-1)-dimensional space we construct a model for these directions by means of concepts characteristic of Euclidean geometry. Because it is a pseudo-orthogonal group, not orthogonal, that operates in this model, the distance between two points and the measure of an angle are not invariants. Using these changeable quantities we construct invariant quantities.

Słowa kluczowe

EN

pseudo-Euclidean geometry; scalar; invariant mapping

PL

geometria pseudoeuklidesowa; geometria nieeuklidesowa; skalar

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2013
• Tom: Vol. 46, nr 2
• Strony: 361--371
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Misiak A.

• School of Mathematics, West Pomeranian University of Technology, Al. Piastów 17, 70-310 Szczecin, Poland

autor

Stasiak E.

autor

Szmuksta-Zawadzka M.

• School of Mathematics, West Pomeranian University of Technology, Al. Piastów 17, 70-310 Szczecin, Poland

Bibliografia

• [1] J. Aczél, S. Gołąb, Functionalgleichungen der Theorie der geometrischen Objekte, P.W.N. Warszawa 1960. Zbl 0100.32901
• [2] L. Bieszk, E. Stasiak, Sur deux formes équivalents de la notion de (r,s)-orientation de la géometrié de Klein, Publ. Math. Debrecen 35 (1988), 43–50.
• [3] B. Glanc, A. Misiak, Z. Stępień, Equivariant mappings from vector product into G-space of vectors and ε-vectors with G = O(n, 1, R), Math. Bohem. 130 (2005), 265–275.
• [4] B. Glanc, A. Misiak, M. Szmuksta-Zawadzka, Equivariant mappings from vector product into G-spaces of φ-scalars with G = O(n – 1,1), Math. Bohem. 132 (2007), 325–332.
• [5] M. Kucharzewski, Über die Grundlagen der Kleinschen Geometrie, Period. Math. Hungar. 8(1) (1977), 83–89. Zbl 0335.50001
• [6] A. Misiak, E. Stasiak, Equivariant maps between certain G-spaces with G = O(n – 1,1), Math. Bohem. 126 (2001), 555–560. Zbl 1031.53031
• [7] A. Misiak, E. Stasiak, G-space of isotropic directions and G-spaces of φ-scalars with G = O(n – 1,1), Math. Bohem. 133 (2008), 289–298.
• [8] E. Stasiak, On a certain action of the pseudoorthogonal group with index one O(n, 1, R) on the sphere Sn2 ( Polish ), Prace Naukowe P. S. 485 (1993).
• [9] E. Stasiak, Scalar concomitants of a system of vectors in pseudo-Euclidean geometry of index 1, Publ. Math. Debrecen 57 (2000), 55–69. Zbl 0966.53012