Induced Pseudo-Riemannian metric on the bundle of orthonormal frames I started reading the book "Foundations of Differential Geometry" from Kobayashi & Nomizu and have a question regarding the possibility to define a metric on the frame bundle in a certain way.
Let $(M,g)$ be a Pseudo-Riemannian manifold, i.e. $g$ is a Pseudo-Riemannian metric on $M$. Let $LM$ be the frame bundle and $OM$ the bundle of $g$-orthonormal frames over $M$.
An isometry $f$ of $M$ can be "lifted" to a diffeomorphism $\tilde{f}$ of the frame bundle, where $\tilde{f}(p):=(Df(p_1),...,Df(p_n))$ for a frame $p\in LM$. Since $f$ is an isometry, $\tilde{f}$ leaves $OM$ invariant. So we can restrict $\tilde{f}$ to a diffeomorphism $\tilde{f}$ of $OM$.
I wonder if it's possible to define a Pseudo-Riemannian metric $\tilde{g}$ on $OM$ such that $\tilde{f}$ is an isometry of $OM$ with respect to that metric. If so, is this metric uniquely determined by the metric $g$ on $M$?
I know that there is a $1:1$ correspondence between metric compatible connections on $(M,g)$ and connection forms on $OM$. Can one use the connection form of the Levi-Civita-connection of $(M,g)$ to define such a metric $\tilde{g}$ on $OM$?
 A: It is possible to define such a metric, but it is not unique and indeed weaker than the structure you get from the Levi-Civita connection. Indeed, let $\theta$ be the usual soldering form on $OM$ (so this has values in $\mathbb R^n$, where $n$ is the dimension of $M$) and let $\gamma$ be the connection form of the Levi-Civita connection. Then $\theta\oplus\gamma$ can be viewed as a one-form on $OM$ with values in $\mathbb R^n\oplus\mathfrak o(p,q)$ (where $(p,q)$ is the signature of the initial pseudo-Riemannian metric). It turns out that this one-form trivializes the tangent bundle of $OM$ and each of the maps $\tilde f$ you describe has the property that $f^*(\theta\oplus\gamma)=\theta\oplus\gamma$. (This is the basis of the description of pseudo-Riemannian metrics as a Cartan geometry.) 
This shows that choosing any inner product on $\mathbb R^n\oplus\mathfrak o(p,q)$ you get a pseudo-Riemannain metric on $OM$ (via the trivialization) for which $f$ is an isometry. Of course there are natrual choices, like the standard inner product of signature $(p,q)$ plus the negative of the Killing form or something like that. However, the property of preserving a trivialization of the tangent bundle is much stronger than the property of being an isometry for a pseudo-Riemannian metric. For example, it is relatively easy to show that the diffeomorphisms preserving a trivialization form a Lie group. 
