Geodesic
On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds
Archivum mathematicum, Tome 50 (2014) no. 5, pp. 287-295 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2014-5-287
Classification : 53B20, 53B30, 53B35, 53B50
Keywords: $F^\varepsilon _2$-planar mapping; $PQ^\varepsilon $-projective equivalence; $F$-planar mapping; fundamental equation; (pseudo-) Riemannian manifold 
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     author = {Hinterleitner, Irena and Mike\v{s}, Josef and Pe\v{s}ka, Patrik},
     title = {On $F^\varepsilon _2$-planar mappings of (pseudo-) {Riemannian} manifolds},
     journal = {Archivum mathematicum},
     pages = {287--295},
     year = {2014},
     volume = {50},
     number = {5},
     doi = {10.5817/AM2014-5-287},
     mrnumber = {3303778},
     zbl = {06487013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2014-5-287/}
}
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Hinterleitner, Irena; Mikeš, Josef; Peška, Patrik. On $F^\varepsilon _2$-planar mappings of (pseudo-) Riemannian manifolds. Archivum mathematicum, Tome 50 (2014) no. 5, pp. 287-295. doi: 10.5817/AM2014-5-287

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