The geodesic curvature $k_g$ can be interpreted as the rate of rotation of the tangent vector about the surface normal, as discussed in this article. The article goes on to say that the integral $\int k_g$ gives the total rotation, which I intuitively understand as the number of times a curve's tangent winds around the normal with respect to the curve itself.
Now, in some physics problems, we are forced to work with an intrinsic picture and so cannot think of a surface normal. While the geometry may be studied in these cases using Christoffel symbols, Gauss-Bonnet, etc, I am lacking an intuitive understanding of what the line integral of the geodesic curvature might mean in terms of rotation of the tangent vector (or some other qualitative picture). Is there such a simple geometric picture?