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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.

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  • $\begingroup$ You may find this answer helpful. It is about the reduction $GL(n)$ to $O(n)$, but if you assume the bundle is orientable, then you can further reduce to $SO(n)$. $\endgroup$ – Michael Albanese Apr 12 at 4:16
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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 $SO(2)$.

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  • $\begingroup$ Can you tell me some reference go the words in your third paragraph about the Riemmanian metric? $\endgroup$ – Danny Apr 13 at 3:11

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