Parallel transport on a parabaloid

Consider a paraboloid z=x^2+y^2. In parametrisation {x=u, y=v, z=u^2+v^2} Christoffel symbols on it should be: Г^1_11 = Г^1_22 = 4u/D Г^2_11 = Г^2_22 = 4v/D all the other are equal to 0. D = 1+4*(u^2+v^2)

Now consider a parallel transport along a curve: u = 0.5*sin(t) v = 0.5*cos(t) t=[0,2*pi] Which must be a circle around the vertex of the paraboloid. Then: Г^1_11 = Г^1_22 = sin(t) Г^2_11 = Г^2_22 = cos(t) The equation for parallel transport of a vector q^i=transpose(q_u q_v): dq^i/dt = -Г^i_jk*q^j*(dx^k/dt) Or: dq_u/dt = -0.5*sin(t)*(q_u*cos(t) - q_v*sin(t)) dq_v/dt = -0.5*cos(t)*(q_u*cos(t) - q_v*sin(t)) I solved the system by odeint from scipy.integrate, here are the results for q^i(0)=transpose(1 0): https://imgur.com/a/h7LUIgS As we can see, at t=2*pi vector didn't match itself at t=0. I know that a vector transported along a closed curve not necessarily match itself, but for such a simple symmetrical case it should, intuitively... Did I something wrong?