Canonical connection form on 2-dim surface and Gaussian Curvature

Let M be a 2-dimensional Riemannian manifold. Let $X_1,X_2$ be an orthonormal frame (w.r.t Riemannian metric) in an open set $U$. Let $(omega_1,omega_2)$ be the dual coframe to $(X_1,X_2)$.

a) Prove that there exists the unique 1-form $omega_2^1$ such that $$domega_1=omega_2^1wedge omega_2 text{ and } domega_2=-omega_2^1wedge omega_1$$

The form $omega^1_2$ is called the canonical connection form of the frame $(X_1, X_2)$.

b) Let $tilde{X_1},tilde{X_2}$ be another orthonormal frame defining the same orientation on $U$. Let $tilde{omega_1}, tilde{omega_2}$ be the corresponding dual coframe. Let $tilde{omega}_2^1$ be the canonical connection form of the frame ($tilde{X}_1,tilde{X}_2$). Prove that $domega_2^1=dtilde{omega}_2^1$

c) Let $K$ and $tilde{K}$ be two functions on U such that
$$domega_2^1=Komega_1wedge omega_2 text{ and } dtilde{omega_2^1}=tilde{K} tilde{omega}_1wedge tilde{omega}_2$$

Show that $K=tilde{K}$.

Solution:

a) Since $omega_1, omega_2$ form a coframe,
$$omega_2^1=f_1omega_1+f_2omega_2, text{ where } f_1=omega_2^1(X_1), f_2=omega_2^1(X_2)$$

$domega_1=omega_2^1wedgeomega_2 implies domega_1(X_1,X_2)=omega_2^1(X_1)$

$domega_2=-omega_2^1wedgeomega_2 implies domega_2(X_1,X_2)=omega_2^1(X_2)$

Therefore $omega_2^1=domega_1(X_1,X_2)omega_1+domega_2(X_1,X_2)omega_2$.

This shows the existence and uniqueness of the canonical connection form. Further, it is easily seen that the conditions are satisfied.

b) Since $X_1,X_2$ is a frame $$tilde{X}_1=a_1X_1+a_2X_2 text{ and } tilde{X}_2=b_1X_1+b_2X_2$$

By orthonormality, $langletilde{X_i},tilde{X}_jrangle=delta_{i,j}$. We get $a_1^2+a_2^2=1=b_1^2+b_2^2$ and $a_1b_1+a_2b_2=0$. Assume $a_1=costheta$ $a_2=sintheta$ and $b_1=sinphi$ $b_2=cosphi$. We get $theta=-phi$. So, $$tilde{X}_1=costheta X_1+sintheta X_2 text{ and } tilde{X}_2=-sin theta X_1+costheta X_2; text{ where } theta text { is a smooth function on U}$$

Similarly, $tilde{omega_i}(tilde{X}_j)=delta_{i,j}$ gives $$tilde{omega}_1=costheta omega_1+sintheta omega_2 text{ and } tilde{omega}_2=-sin theta omega_1+costheta omega_2; $$

I don’t know how to continue further. Any help is appreciated.

Thanks