Exercise:
Find the geodesic curvature of the parallel $\gamma (t) = (a\cos t, a\sin t, r) $ of $T= \{ (x,y,z) \in \mathbb{R^3} \mid \big(\sqrt{x^2+y^2} -a^2 \big)^2+z^2 = r^2\}$, with $a>r>0$ and $t\in ]0,2 \pi[$.
My attempt:
I used this parametrisation of $T$:
$$X(u_1,u_2) = \big((r\cos u_1+a)\cos u_2,\; (r\cos u_1+a)\sin u_2,\; r\sin u_1\big)$$
with $u_1, u_2\in ]0,2 \pi[$.
For $X$, I have $N= -(\cos u_1\cos u_2,\; \cos u_1\sin u_2,\; \sin u_1)$.
I don't know how I should find $N$ for the parallel $\gamma(t)$. If I have $N$, I know how to compute the geodesic curvature, but I don't understand how I should evaluate $N$ in this case.