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

Oldřich Kowalski and Masami Sekizawa

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


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