Let $$M$$ be a surface with Riemannian metric $$g$$. Recall that an orthonormal framing of $$M$$ is an ordered pair of vector fields $$(E_1,E_2)$$ such that $$g(E_i,E_j)=\delta_{ij}$$. Prove that an orthonormal framing exists iff $$M$$ is orientable and $$M$$ admits a nowhere vanishing vector field $$X$$.(adsbygoogle = window.adsbygoogle || []).push({});

Remark: It's obvious in $$\mathbb{R}^3$$, but how to formally justify it?

The definition for orientabily: $$M$$ is orientable if there exists an atlas $$(u_{\alpha},M_{\alpha})_{\alpha}$$ such that $$\mathrm{det}(\mathrm{d}(u_{\beta}\circ u_{\alpha}^{-1}))>0$$, for each $$(\alpha,\beta)$$ such that $$M_{\alpha} \cap M_{\beta} \neq \Phi$$

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# Existence of orthonormal frame

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