Geodesic
Invariance of $g$-natural metrics on linear frame bundles
Archivum mathematicum, Tome 44 (2008) no. 2, pp. 139-147.

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In this paper we prove that each $g$-natural metric on a linear frame bundle $LM$ over a Riemannian manifold $(M, g)$ is invariant with respect to a lifted map of a (local) isometry of the base manifold. Then we define $g$-natural metrics on the orthonormal frame bundle $OM$ and we prove the same invariance result as above for $OM$. Hence we see that, over a space $(M, g)$ of constant sectional curvature, the bundle $OM$ with an arbitrary $g$-natural metric $\tilde{G}$ is locally homogeneous.
Classification : 53C07, 53C20, 53C21, 53C40
Keywords: Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity 
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     author = {Kowalski, Old\v{r}ich and Sekizawa, Masami},
     title = {Invariance of $g$-natural metrics on linear frame bundles},
     journal = {Archivum mathematicum},
     pages = {139--147},
     publisher = {mathdoc},
     volume = {44},
     number = {2},
     year = {2008},
     mrnumber = {2432851},
     zbl = {1212.53042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2008__44_2_a6/}
}
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Kowalski, Oldřich; Sekizawa, Masami. Invariance of $g$-natural metrics on linear frame bundles. Archivum mathematicum, Tome 44 (2008) no. 2, pp. 139-147. http://geodesic.mathdoc.fr/item/ARM_2008__44_2_a6/