I have to prove that "any straight line $\alpha$ contained on a surface $S$ is an asymptotic curve and geodesic (modulo parametrization) of that surface $S$". Can I have hints at tackling this problem? It seems so general that I am not sure even how to formulate it well, let alone prove it. Intuitively, I imagine that the normal $n_{\alpha}$ to the line/curve is perpendicular to the normal vector $N_{S}$ to the surface $S$, thus resulting in the asymptoticity; alternatively, a straight line has curvature $k = 0$ everywhere, and so the result follows. Is this reasoning adequate for a proof of the first part? I also realize that both geodesics and straight lines are the paths of shortest distance between two points on given surfaces (here, both $S$), thus the straight line must be a geodesic of any surface which contains it; should I quantify this statement, though?
Let $\mathbb{H}^2 = \{(x, y) \in \mathbb{R}^2: y>0 \}$ be the hyperbolic plane with the Riemannian metric $(\mathbb{d}s)^2 =\frac{(\mathbb{d}x)^2+(\mathbb{d}y)^2}{y^2}$. Consider a "square" $P = \{ (x, y) \in \mathbb{H}^2: 1 \leq x,y \leq 2 \}$. I need to calculate the geodesic curvature of the sides of $P$ and, for Gaussian curvature $K$ of $P$, I have to calculate $\int_{P} (K) \mathbb{d}\sigma$, where $\mathbb{d}\sigma$ is the area measure element of $\mathbb{H}^2$. Just hints as to how to start would be helpful. (I see that I have the first fundamental form, from which I can derive the coefficients $E$, $F$, and $G$ and thereby (hopefully easily) Christoffel symbols and an expression for area, but I do not see how any of this takes the actual square into account. Only the coordinates at which I evaluate these quantities seem to come come from the square! But I would still like detailed examples of even these things, please.)
Best Answer
Intuitively, I imagine that the normal $n_{\alpha}$ to the line/curve is perpendicular to the normal vector $N_{S}$ to the surface $S$, thus resulting in the asymptoticity; alternatively, a straight line has curvature $k = 0$ everywhere, and so the first result follows.
I also realize that both geodesics and straight lines are the paths of shortest distance between two points on given surfaces (here, both $S$), thus the straight line must be a geodesic of any surface which contains it - again, here $S$.