The pushforward of transition function of an orientable Riemannian manifold is in $SO(n)$?

Question

Based on this notes last line, the pushforward of transition function of an orientable Riemannian manifold of dimension $2$ is in $SO(2)$. I wonder why it is true.

Answer 1

Let $F(M) \to M$ be the frame bundle of $M$, that is,

$$F(M) = \bigcup_{p \in M} \{ (v_1, v_2) \in T_p M \times T_p M : \{v_1, v_2 \} \text{ is a basis of } T_p M \}.$$

Then, setting $\pi : F(M) \to M$, $\pi((v_1, v_2)) = p$ if $v_i \in T_p M$, we have a principal bundle with structure group $GL(2)$.

A riemmanian metric on $M$ is the choice of a reduction of the structure group of this bundle to $O(2)$. The manifold is orientable iff the structure group can be further reduced to $SO(2)$.

Complement: the push forward of the transition function that appears in the notes takes an orthonormal frame to another, so it must belong to $O(2)$. Since the manifold is oriented, the Jacobian of the transition function is positive. Thus, it belongs to $SO(2)$.