Publication result detail

On $F^\varepsilon_2$-planar mappings of (pseudo-) Riemannian manifolds

HINTERLEITNER, I.; MIKEŠ, J.; PEŠKA, P.

Original Title

On $F^\varepsilon_2$-planar mappings of (pseudo-) Riemannian manifolds

English Title

On $F^\varepsilon_2$-planar mappings of (pseudo-) Riemannian manifolds

Type

Scopus Article

Original Abstract

We study special F-planar mappings between two n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced $PQ^{\varepsilon}$-projectivity of Riemannian metrics, $\varepsilon\neq 1,1+n$. Later these mappings were studied by Matveev and Rosemann. They found that for $\varepsilon=0$ they are projective. We show that $PQ^{\varepsilon}$-projective equivalence corresponds to a special case of F-planar mapping studied by Mikeš and Sinyukov (1983) and ${F_2}$-planar mappings (Mikeš, 1994), with F=Q. Moreover, the tensor P is derived from the tensor Q and the non-zero number $\varepsilon$. For this reason we suggest to rename $PQ^{\varepsilon}$ as ${F_2^{\varepsilon}}$. We use earlier results derived for F- and $F_2$-planar mappings and find new results. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.

English abstract

We study special F-planar mappings between two n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced $PQ^{\varepsilon}$-projectivity of Riemannian metrics, $\varepsilon\neq 1,1+n$. Later these mappings were studied by Matveev and Rosemann. They found that for $\varepsilon=0$ they are projective. We show that $PQ^{\varepsilon}$-projective equivalence corresponds to a special case of F-planar mapping studied by Mikeš and Sinyukov (1983) and ${F_2}$-planar mappings (Mikeš, 1994), with F=Q. Moreover, the tensor P is derived from the tensor Q and the non-zero number $\varepsilon$. For this reason we suggest to rename $PQ^{\varepsilon}$ as ${F_2^{\varepsilon}}$. We use earlier results derived for F- and $F_2$-planar mappings and find new results. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.

Keywords

$F^\varepsilon_2$-planar mapping; $PQ^\varepsilon$-projective equivalence; F-planar mapping; fundamental equation; (pseudo-) Riemannian manifold.

Key words in English

$F^\varepsilon_2$-planar mapping; $PQ^\varepsilon$-projective equivalence; F-planar mapping; fundamental equation; (pseudo-) Riemannian manifold.

Authors

HINTERLEITNER, I.; MIKEŠ, J.; PEŠKA, P.

RIV year

2015

Released

19.12.2014

Publisher

Masaryk University

Location

Brno, Czech Republic

ISBN

0044-8753

Periodical

ARCHIVUM MATHEMATICUM

Volume

50

Number

5

State

Czech Republic

Pages from

33

Pages to

41

Pages count

9

BibTex

@article{BUT110152,
  author="Irena {Hinterleitner} and Josef {Mikeš} and Patrik {Peška}",
  title="On $F^\varepsilon_2$-planar mappings of (pseudo-) Riemannian manifolds",
  journal="ARCHIVUM MATHEMATICUM",
  year="2014",
  volume="50",
  number="5",
  pages="33--41",
  doi="10.5817/AM2014-5-287",
  issn="0044-8753"
}