# On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds

## Irena Hinterleitner, Josef Mikeš, and Patrik Peška

Address:

Irena Hinterleitner, Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics, Žižkova 17, 602 00 Brno, Czech Republic

Josef Mikeš, Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic

Patrik Peška, Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 77146 Olomouc, Czech Republic

E-mail:

hinterleitner.irena@seznam.cz

josef.mikes@upol.cz

patrik_peska@seznam.cz

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.

AMSclassification: primary 53B20; secondary 53B30, 53B35, 53B50.

Keywords: F^\varepsilon _2-planar mapping, PQ^\varepsilon -projective equivalence, F-planar mapping, fundamental equation, (pseudo-) Riemannian manifold.

DOI: 10.5817/AM2014-5-287