Metric geometries in an axiomatic perspective

Victor Pambuccian, Horst Struve, Rolf Struve

Research output: Chapter in Book/Report/Conference proceedingChapter

2 Scopus citations

Abstract

In his 1854 Habilitationsvortrag Riemann presented a new concept of space endowed with a metric of great generality, which, through specification of the metric, gave rise to the spaces of constant curvature. In a different vein, yet with a similar aim, J. Hjelmslev, A. Schmidt, and F. Bachmann, introduced axiomatically a very general notion of plane geometry, which provides the foundation for the elementary versions of the geometries of spaces of constant curvature. We present a survey of these absolute geometric structures and their first-order axiomatizations, as well as of higher-dimensional variants thereof. In the 2-dimensional case, these structures were called metric planes by F. Bachmann, and they can be seen as the common substratum for the classical plane geometries: Euclidean, hyperbolic, and elliptic. They are endowed with a very general notion of orthogonality or reflection that can be specialized into that of the classical geometries by means of additional axioms. By looking at all the possible ways in which orthogonality can be introduced in terms of polarities, defined on (the intervals of a chain of subspaces of) projective spaces, one obtains a further generalization: theCayley-Klein geometries.We present a survey of projective spaces endowed with an orthogonality and the associated Cayley-Klein geometries.

Original languageEnglish (US)
Title of host publicationFrom Riemann to Differential Geometry and Relativity
PublisherSpringer International Publishing
Pages413-455
Number of pages43
ISBN (Electronic)9783319600390
ISBN (Print)9783319600383
DOIs
StatePublished - Jan 1 2017

ASJC Scopus subject areas

  • Mathematics(all)

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