Total Curvature Defined with Respect to Some Arbitrary Geodesic Curvature

Question

If $\gamma$ is a smooth curve in $R^2$, length parametrized with respect to Euclidean metric, then the integral of $|\ddot{\gamma}|$ over the curve gives the amount the tangent vector rotates (if it is a closed immersed curve you get $2\pi k$ for some itneger $k$). Now instead of the Euclidean curvature, if you use the geodesic curvature of the curve with respect to some metric $g$ (assuming now the curve is length parametrized with respect to this metric) you get that the geodesic curvature is $\dot{\phi} + \omega(\dot{\gamma})$, where $\omega$ is the connection 1-form with respect to some orthogonal basis $E_1,E_2$ and $\dot{\gamma}=cos(\phi)E_1 + sin(\phi)E_2$. Now if you integrate this quantity over the curve, you get something which equals to amount of rotaion of the tangent vector + integral of the connection 1-form evaluated on $\dot{\gamma}$. So what does this second part of the integral geometrically mean?

Answer 1

It is telling you the rate at which the moving frame $E_1,E_2$ is twisting from being parallel ("fixed") along the curve. In particular, it vanishes when $E_1$ is parallel along the curve.