# Invariance of $g$-natural metrics on linear frame bundles

## Oldřich Kowalski and Masami Sekizawa

Address:

Faculty of Mathematics and Physics, Charles University in Prague Sokolovská 83, 186 75 Praha 8, Czech Republic

Department of Mathematics, Tokyo Gakugei University Koganei-shi Nukuikita-machi 4-1-1, Tokyo 184-8501, Japan

E-mail:

kowalski@karlin.mff.cuni.cz

sekizawa@u-gakugei.ac.jp

Abstract: 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.

AMSclassification: Primary: 53C07; Secondary: 53C20, 53C21, 53C40.

Keywords: Riemannian manifold, linear frame bundle, orthonormal frame bundle, $g$-natural metrics, homogeneity