I've worked through a problem my professor gave me on normal/geodesic curvature. I believe there was an error in his solution because in his answer he states the geodesic curvature in the following problem is zero, which I believe is wrong.
The elliptic paraboloid $S: z = x^2 + y^2$ is parametrized by $\sigma(u,v) = (u\cos(v), u\sin(v), u^2)$.
a) Find the second fundamental form of $\sigma$.
Solution:
$\mathrm{I\!I} = L\mathit{du}^2 + 2M\mathit{du dv} + N\mathit{dv}^2$
$n = \frac{\sigma_u \times \sigma_v}{||\sigma_u \times \sigma_v||} = \frac{1}{\sqrt{4u^2 + 1}}(-2u\cos(v),-2u\sin(v),1)$
$L = \sigma_{uu} \cdot n = \frac{2}{\sqrt{4u^2 + 1}}$
$M = \sigma_{uv} \cdot n = 0$
$N = \sigma_{vv} \cdot n = \frac{2u^2}{\sqrt{4u^2 + 1}}$
$\mathrm{I\!I} = \frac{2}{\sqrt{4u^2 + 1}} \mathit{du^2} + \frac{2u^2}{\sqrt{4u^2 + 1}} \mathit{dv^2}$
b) Find the normal and geodesic curvatures of the circle $x^2 + y^2 = 4$ on $S$.
Solution:
$\gamma(t) = (2\cos(t), 2\sin(t), 4)$
$\dot{\gamma}(t) = (-2\sin(v), 2\cos(t), 0)$
$\ddot{\gamma}(t) = (-2\cos(v), -2\sin(v), 0)$
Find unit-speed reparametrization:
$c(s) = (2\cos(\frac{t}{2}), 2\sin(\frac{t}{2}, 4)$
$\dot{c}(s) = (-\sin(\frac{s}{2}), \cos(\frac{s}{2}), 0)$
$\ddot{c}(s) = (-\frac{1}{2}\cos(\frac{s}{2}), -\frac{1}{2}\sin(\frac{s}{2}), 0)$
$n(c(s)) = \frac{1}{\sqrt{4(2^2) + 1}}(-4\cos(\frac{s}{2}), -4\sin(\frac{s}{2}), 1)$
Using $K_n = \ddot{\gamma}(t) \cdot n(\gamma(t))$,
$K_n = \frac{2}{\sqrt{17}}$, but since $x^2 + y^2 = 4$ is ultimately a circle of radius 2, its curvature K should be $\frac{1}{2}$. In the notes, the geodesic curvature is said to be 0, but when doing $K_g = \ddot{\gamma}(t) \cdot (n \times \dot{\gamma} (t))$, I get $K_g = \frac{1}{2\sqrt{17}}$. And when checking with the property $K^2 = K_n^2 + K_g^2$, $K_g$ should not be zero. My question is if $K_g$ is indeed not zero, what does it mean for this circle to have geodesic curvature? Does the curvature only exist because it lies on the surface? If this circle were on a cylinder of radius 2 would it still have the same geodesic curvature?