Cayley-Klein Spaces

Alexander I. Bobenko; Carl O.R. Lutz; Helmut Pottmann; Jan Techter

doi:10.1007/978-3-030-81847-0_4

Cayley-Klein Spaces

Alexander I. Bobenko^*, Carl O.R. Lutz, Helmut Pottmann, Jan Techter

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

In Klein’s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and elliptic space can be obtained by using a quadric to induce the corresponding metric [Kle1928]. In this section we introduce the corresponding general notion of Cayley-Klein spaces and their groups of isometries, see, e.g., [Kle1928, Bla1954, Gie1982]. We put a particular emphasis on the description of hyperplanes, hyperspheres, and their mutual relations.

Original language	English (US)
Title of host publication	SpringerBriefs in Mathematics
Publisher	Springer Science and Business Media B.V.
Pages	27-36
Number of pages	10
DOIs	https://doi.org/10.1007/978-3-030-81847-0_4
State	Published - 2021

Publication series

Name	SpringerBriefs in Mathematics
ISSN (Print)	2191-8198
ISSN (Electronic)	2191-8201

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

ASJC Scopus subject areas

General Mathematics

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10.1007/978-3-030-81847-0_4

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title = "Cayley-Klein Spaces",

abstract = "In Klein{\textquoteright}s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and elliptic space can be obtained by using a quadric to induce the corresponding metric [Kle1928]. In this section we introduce the corresponding general notion of Cayley-Klein spaces and their groups of isometries, see, e.g., [Kle1928, Bla1954, Gie1982]. We put a particular emphasis on the description of hyperplanes, hyperspheres, and their mutual relations.",

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AU - Bobenko, Alexander I.

AU - Lutz, Carl O.R.

AU - Pottmann, Helmut

AU - Techter, Jan

PY - 2021

Y1 - 2021

N2 - In Klein’s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and elliptic space can be obtained by using a quadric to induce the corresponding metric [Kle1928]. In this section we introduce the corresponding general notion of Cayley-Klein spaces and their groups of isometries, see, e.g., [Kle1928, Bla1954, Gie1982]. We put a particular emphasis on the description of hyperplanes, hyperspheres, and their mutual relations.

AB - In Klein’s Erlangen program Euclidean and non-Euclidean geometries are considered as subgeometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and elliptic space can be obtained by using a quadric to induce the corresponding metric [Kle1928]. In this section we introduce the corresponding general notion of Cayley-Klein spaces and their groups of isometries, see, e.g., [Kle1928, Bla1954, Gie1982]. We put a particular emphasis on the description of hyperplanes, hyperspheres, and their mutual relations.

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M3 - Chapter

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PB - Springer Science and Business Media B.V.

ER -

Cayley-Klein Spaces

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