I'm trying to learn by myself differential geometry by the Do Carmo book.
However, there isn't any example on how to solve this type of exercise on trying to found what parallels of a certain surface are geodesics anywere in the book, only theory and I've struggling without any sucess. Here is an example.
What parallels of the this torus are geodesics? $$X(u,v)=((a+r*cos(u))*cos(v) , (a+r*cos(u))*sin(v), r*sin(u))$$
Please, can someone tell me how do I solve this type of problems properly? I just need one example to get the pattern. Thanks.
Use the fact that $\alpha(t) = \mathbf{x}(u(t), v(t)) \subset S$ is a geodesic if and only if $$ u'' E + v'' F = -(u')^2 E_u /2 - u'v'E_v - (v')^2 \Big( F_v - G_u/2 \Big) $$ and $$ v''G + u''F = -(v')^2G_v /2 - u'v' G_u - (u')^2 \Big(F_u - E_v/2\Big), $$ where $E, F, G$ are the coefficients of the first fundamental form. For the case of a torus parametrized by $$ \mathbf{x}(u,v) = ((R+r \cos v) \cos u, (R + r \cos v) \sin u, r \sin v) $$ (note this is slightly different from your parametrization just because I already have these equations worked out) this translates into solving the equations $$ u''(t) = \frac{2ru'(t)v'(t) \sin v(t)}{R+r\cos v(t)}$$ and $$ v''(t) = - \frac{(R+r \cos v(t))(u'(t))^2 \sin v(t)}{r} $$ from which you should be able to find the parallels which are geodesics. From intuition, I believe they should be the two circles $((R-r)\cos u, (R-r)\sin u, 0)$ and $((R+r) \cos u , (R+r) \sin u, 0)$.