On F 2 ε -planar mappings of (pseudo-) Riemannian manifolds

Irena Hinterleitner; Josef Mikeš; Patrik Peška

Archivum Mathematicum (2014)

  • Volume: 050, Issue: 5, page 287-295
  • ISSN: 0044-8753

Abstract

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We study special F -planar mappings between two n -dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced P Q ε -projectivity of Riemannian metrics, ε ≠ 1 , 1 + n . Later these mappings were studied by Matveev and Rosemann. They found that for ε = 0 they are projective. We show that P Q ε -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 ε . For this reason we suggest to rename P Q ε as F 2 ε . 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.

How to cite

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Hinterleitner, Irena, Mikeš, Josef, and Peška, Patrik. "On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds." Archivum Mathematicum 050.5 (2014): 287-295. <http://eudml.org/doc/262140>.

@article{Hinterleitner2014,
abstract = {We study special $F$-planar mappings between two $n$-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced $PQ^\{\varepsilon \}$-projectivity of Riemannian metrics, $\varepsilon \ne 1,1+n$. 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.},
author = {Hinterleitner, Irena, Mikeš, Josef, Peška, Patrik},
journal = {Archivum Mathematicum},
keywords = {$F^\varepsilon _2$-planar mapping; $PQ^\varepsilon $-projective equivalence; $F$-planar mapping; fundamental equation; (pseudo-) Riemannian manifold; $F^\varepsilon _2$-planar mapping; $PQ^\varepsilon $-projective equivalence; -planar mapping; fundamental equation; (pseudo-)Riemannian manifold},
language = {eng},
number = {5},
pages = {287-295},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds},
url = {http://eudml.org/doc/262140},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Hinterleitner, Irena
AU - Mikeš, Josef
AU - Peška, Patrik
TI - On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 5
SP - 287
EP - 295
AB - We study special $F$-planar mappings between two $n$-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced $PQ^{\varepsilon }$-projectivity of Riemannian metrics, $\varepsilon \ne 1,1+n$. 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.
LA - eng
KW - $F^\varepsilon _2$-planar mapping; $PQ^\varepsilon $-projective equivalence; $F$-planar mapping; fundamental equation; (pseudo-) Riemannian manifold; $F^\varepsilon _2$-planar mapping; $PQ^\varepsilon $-projective equivalence; -planar mapping; fundamental equation; (pseudo-)Riemannian manifold
UR - http://eudml.org/doc/262140
ER -

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