Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$
Calculate the geodesic curvature of $\gamma$.
The problem I'm having is that every way I know of calculating the geodesic curvature of a curve in a surface involves knowledge of a normal vector and I'm not sure how one would go about defining the tangential derivative for such abstract smooth surfaces. Any clarification would be appreciated.
Best Answer
HINT: You should have a formula for geodesic curvature just in terms of the first fundamental form (and the curve, of course) if you're working in an orthogonal parametrization. (You don't say what material you know and what tools you have at your disposal. It's also a very straightforward computation using differential forms and moving frames.)