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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; $F$-planar mapping; fundamental equation; (pseudo-) Riemannian manifold
Summary:
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.
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