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Keywords:
Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity
Riemannian manifold; linear frame bundle; orthonormal frame bundle; $g$-natural metrics; homogeneity
Summary:
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.
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