On F_2^ε -Planar Mappings with Function ε of (pseudo-) Riemannian Manifolds

Hana Chudá, Naděžda Guseva, Patrik Peška


In this paper we study special mappings between $n$-dimensional (pseudo-) Riemannian manifolds. In $2003$ Topalov introduced $PQ^\varepsilon$- projectivity of Riemannian metrics, with constant $\varepsilon \neq 0, 1 + n$. These mappings were studied later by Matveev and Rosemann and they found that for $\varepsilon = 0$ they are projective.
These mappings could be generalized for case, when $\varepsilon$ will be a function on manifold.
We show that $PQ^\varepsilon$ - projective equivalence with $\varepsilon$ is a function corresponds to a special case of $F$-planar mapping, studied by Mikes and Sinyukov $(1983)$ with $F=Q$. Moreover, the tensor $P$ is derived from the tensor $Q$ and non-zero function $\varepsilon$.

We assume that studied mappings will be also $F_2$- planar (Mike\v s 1994). This is the reason, why we suggest to rename $PQ^\varepsilon$ mapping as $F^{\varepsilon}_{2}$.
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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