Prove that any smooth, orientable, and connected surface can be equipped with a Riemann surface structure (in the sense of a complex 1-dimensional manifold).
My idea is to modify the chart so that the transition map satisfies the Cauchy-Riemann equation and is thus holomorphic. Since the manifold is orientable, the jacobian of the transition maps everywhere is invertible. And say let $f = \phi_\alpha \circ \phi_\beta^{-1}$ for some transition maps on $U \subset R^2$. I want to compose each $\phi_\alpha$ with some $\psi_\alpha$ where $$J_{\psi_\alpha \circ \phi_\alpha \circ \phi_\beta^{-1}\circ \psi_\beta^{-1}}(\psi_\beta^{-1}(p))= J_{\psi_\alpha}(f(p))J_f(p)J_{\psi_\beta^{-1}}(\psi_\beta^{-1}(p))$$
such that $$J_{\psi_\alpha}(f(p)) = \begin{bmatrix} a &b \\ b & -a \end{bmatrix}^{\frac{1}{2}}\begin{bmatrix} \partial_xf_1(p) &\partial_yf_1(p) \\ \partial_xf_2(p) & \partial_yf_2(p) \end{bmatrix}^{-\frac{1}{2}}$$
and $$J_{\psi_\beta^{-1}}(\psi_\beta^{-1}(p)) = \begin{bmatrix} \partial_xf_1(p) &\partial_yf_1(p) \\ \partial_xf_2(p) & \partial_yf_2(p) \end{bmatrix}^{-\frac{1}{2}}\begin{bmatrix} a &b \\ b & -a \end{bmatrix}^{\frac{1}{2}}$$ for some $a,b$ properly chosen.
If such construction exists, then the jacobian for the new transition map is $\begin{bmatrix} a &b \\ b & -a \end{bmatrix}$, satisfying the Cuacy-Riemann equation and thus the transition holomorphic. Does such construction work and does there exist such $\psi_\alpha$? I think properly not because I didn't use the connectedness property. So is there any modification I can do to make this construction valid?
