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

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.

• 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 $SO(n)$. – Michael Albanese Apr 12 at 4:16

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 $$SO(2)$$.