Why the geodesic curvature is invariant under isometric transformations? As I know the geodesic curvature
$$
\kappa_g = \sqrt{\det~g} \begin{vmatrix}  \frac{du^1}{ds} & \frac{d^2u^1}{ds^2} + \Gamma^1_{\alpha\beta}  \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \\  \frac{du^2}{ds} & \frac{d^2u^2}{ds^2} + \Gamma^2_{\alpha\beta}  \frac{du^\alpha}{ds} \frac{du^\beta}{ds} \end{vmatrix}, 
$$
where $g$ is the metric tensor, $\Gamma^v_{\alpha\beta}$ is the Christoffel symbols of the second kind.
And the first fundamental form of the surface $I = (du^1, du^2) g (du^1, du^2)^T$. I think $I$ is invariant under isometric transformations but not the metric tensor $g$. So why $\kappa_g$ is invariant under isometric transformations?
 A: $ \kappa_g$ depends purely on the coefficients of the first fundamental form ( of surface theory FFF) and their derivatives, second fundamental form SFF coefficients are not involved.
It is invariant in isometric mappings ( bending transformations) like lengths,angles,  $K$ Gauss curvature , integral curvature etc. Liouville's theorem gives the expressions. Reference of text books of Differential geometry.
$K$ is an exception where the determinants of SFF and FFF can be used to  derive it in  the Gauss Egregium theorem.
A: 
This is a well known sequence of isometric transformations of the (left-handed) Helicoid into the Catenoid back into the (right_handed) Helicoid. I've put in examples of the  principal lines of curvature,  which are both geodesics . The drawing isn't perfect , the surfaces lying on the right and left sides are missing a red straight line ( a geodesic)  down the center line of the helicoid. (Compare the central line on the Catenoid -- a circle ), from edge on the blue lines of the Helicoid are also straight lines.  Incidently the drawing is quite large, you can get a better idea of it by taking it down off of your screen. 
A reference for $\kappa_g$ is "Lectures on Classical Differential Geometry" by Dirk J. Struik ( Chapter Four , Geometry on the Surface ,
pp. 128 ). It is shown that geodesic curvature depends only on E,F,G hence it is an invariant under isometric transformations, or a bending invariant as it is called in the text. 
