Let $F(M)$ be the first order frame bundle on a $C^\infty$ smooth real manifold $M$ of dimension $m$, and $O(M)$ the bundle of all orthonormal frames. $F(M)$ is a $GL(m)$-principal bundle, and $O(M)$ is an $O(m)$-principal bundle.
In the following, I use $G$ to denote either $GL(m)$ or $O(m)$ and $P$ to denote the total space $F(M)$ or $O(M)$ respectively, of the principal fibre bundle $(F(M),p,M,GL(m))$ or $(O(M),p,M,O(m))$, respectively. So the general notation of the PFB is
$(P,p,M,G)$.
Let $V$ be a real vector space and $\rho: G \to GL(V)$ a linear representation of $G$ on $V$. A smooth map $f: P \to V$ is called a "$G$-tensor of type $(V,\rho)$ on $M$," if
$\forall g \in G, u \in P: f(u.g) = \rho(g^{-1})(f(u))$
Now for my question:
Does every $O(m)$-tensor of type $(V,\rho)$ on $M$ lift to a $GL(m)$-tensor of type $(V\!\uparrow_{O(m)}^{GL(m)}\, , \rho\!\uparrow_{O(m)}^{GL(m)}\, )$ for the induced representation $\rho\!\uparrow_{O(m)}^{GL(m)}$ of the group $GL(m)$ on $V$?
In other words, does every Cartensian (Euclidean) tensor field on $M$ lift to an affine tensor field on $M$?
My guess is: Yes. Somehow, because $O(m)$ is the maximal compact subgroup of $GL(m)$. But is there an "easy" proof, or a reference to the literature? Thank you very much!
Remark: Maybe just consider the associated bundles, instead of sections, and use an extension of structure group: Let $E$ be the associated vector bundle to $(O(M),p,M,O(m))$ with typical fibre $V$ obtained from the action $\rho$ of $O(m)$ on $V$. Given the group homomorphism (embedding) $\iota: O(m) \hookrightarrow GL(m)$, apply the "extension of structure groups functor" $GL(M) \times_{M,\iota, O(m)} \, (.) $ to the bundle $E$, and then prove that $GL(M) \times_{M,\iota, O(m)} E $ is isomorphic, as real vector bundles over $M$, to $E$?
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Edit 1, one day later, due to some findings: The following theorem can be found in a compilatory master thesis by Sandon:
So, if the left action $\rho: O(m) \to Aut(V)$ on $V$ is the restriction to $O(m)$ of a left action $GL(m) \to Aut(V)$ on the same vector space $V$, then, indeed, the (associated) vector bundles (over $M$) $F(M) \times_{GL(m)} V$ and $O(M) \times_{O(m)} V$ are isomorphic (as vector bundles over $M$), and have corresponding smooth sections.
Therefore, for such $O(m)$-actions $(\rho\downarrow^{Gl(m)}_{O(m)} \, ,V)$, which are a restriction of a $GL(m)$ action $\rho$, every $O(m)$-tensor of type $(\rho\downarrow^{Gl(m)}_{O(m)},V)$ indeed stems from a $GL(m)$-tensor of type $(\rho,V)$, Cartensian and affine tensors are the same.
I guess, the general case can now be settled by using Frobenius reciprocity, universal property of the induced action, and/or Mackey machinery. Because a given left $O(m)$ action can be considered the restriction to $O(m)$ of its $GL(m)$-induction.