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.
@article{116931,
author = {Old\v rich Kowalski and Masami Sekizawa},
title = {Invariance of $g$-natural metrics on linear frame bundles},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {139-147},
zbl = {1212.53042},
mrnumber = {2432851},
language = {en},
url = {http://dml.mathdoc.fr/item/116931}
}
Kowalski, Oldřich; Sekizawa, Masami. Invariance of $g$-natural metrics on linear frame bundles. Archivum Mathematicum, Tome 044 (2008) pp. 139-147. http://gdmltest.u-ga.fr/item/116931/
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