Reflection-Like Maps in High-Dimensional Euclidean Space

Zhiyong Huang and Baokui Li
Additional contact information
Zhiyong Huang: School of Mathematics, Renmin University of China, Beijing 100872, China
Baokui Li: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Mathematics, 2020, vol. 8, issue 6, 1-11

Abstract: In this paper, we introduce reflection-like maps in n -dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

Keywords: line-to-line maps; reflection-like maps; affine transformations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/6/872/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/6/872/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:6:p:872-:d:364342

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().