Instructions
This notebook does exactly one thing: give you a cute visualization of parallel transport along a circle of latitude on a sphere.  There’s no code for you to add or edit; just compile the cell below.

The resulting manipulate will depict a sphere with a circle of latitude at latitude (Subscript[ϕ, 0]), a basis {(Subscript[E, 1]),(Subscript[E, 2])} at the point α(t) on the circle of latitude (this basis is not parallel transported), and a vector V(t) which is parallel transported along the curve, having made an angle of (Subscript[θ, 0]) with (Subscript[E, 1]) at t=0.

The manipulate will have three sliders: you may control tt, the parameter for the circle of latitude; (Subscript[θ, 0]), the initial angle between V and (Subscript[E, 1]); and (Subscript[ϕ, 0]), the latitude of the circle.

Clear[RR];
Clear[ϕ0];
RR=3;
surfaceX[u1_,u2_]:={RR*Cos[u2]*Cos[u1],RR*Cos[u2]*Sin[u1],RR*Sin[u2]};
αU[t_,ϕ0_]:={t,ϕ0};
α[t_,ϕ0_]:=surfaceX[Part[αU[t,ϕ0],1],Part[αU[t,ϕ0],2]];
x1[u1_,u2_]:=D[surfaceX[uu1,u2],uu1]/.uu1->u1;
x2[u1_,u2_]:=D[surfaceX[u1,uu2],uu2]/.uu2->u2;
e1[u1_,u2_]:=((x1[u1,u2])/(Norm[x1[u1,u2]]));
e2[u1_,u2_]:=((x2[u1,u2])/(Norm[x2[u1,u2]]));
v[t_,θ0_,ϕ0_]:=Cos[θ0-t*Sin[ϕ0]]*e1[Part[αU[t,ϕ0],1],Part[αU[t,ϕ0],2]] + Sin[θ0-t*Sin[ϕ0]]*e2[Part[αU[t,ϕ0],1],Part[αU[t,ϕ0],2]];

plot={
ParametricPlot3D[surfaceX[u1,u2],{u1,-Pi,Pi},{u2,-Pi/2,Pi/2},PlotStyle->Opacity[0.5],Mesh->None]
};
Manipulate[Show[
plot,
ParametricPlot3D[α[t,ϕ0],{t,0,2Pi}],
Graphics3D[{Black,Arrow[{α[tt,ϕ0],α[tt,ϕ0]+e1[Part[αU[tt,ϕ0],1],Part[αU[tt,ϕ0],2]]}]}],
Graphics3D[{Black,Arrow[{α[tt,ϕ0],α[tt,ϕ0]+e2[Part[αU[tt,ϕ0],1],Part[αU[tt,ϕ0],2]]}]}],
Graphics3D[{Red,Arrow[{α[tt,ϕ0],α[tt,ϕ0]+v[tt,θ0,ϕ0]}]}]
],{{tt,0},0,2Pi},{θ0,0,2Pi},{{ϕ0,0},-Pi/2,Pi/2}]