I need to solve the following exercise. I wonder whether my solution is correct.
Problem: Take a sphere in $\mathbb{R}^3$ centered around the origin of radius $R$. Consider the spherical triangle $ABC$: $A=(0,0,R),\ B=(R,0,0),\ C=(R \cos(\alpha),R\sin(\alpha),0)$. Find the image of the vector $V_1=(1,0,0)$ in $A$ under the parallel transport along the edges of the triangle.
My solution:
I want to find the image of the vector $V_1=(1,0,0)$ in $A$ under the parallel transport along the edges of the triangle. My calculations showed that the parallel transport takes the vector $V$ in $A$ (throught the "spherical" edge $AB$ ) to the vector $V_2=(0,0,-1)$ in $B$. Similarly the parallel transport through $BC$ takes $V_2$ to $V_2$ in $C$. The parallel transport of $V_2$ throught $CA$ is $(\cos(\alpha),\sin(\alpha),0)$.
So the parallel transport of $V_1$ in $A$ along the edges of $ABC$ is $(\cos(\alpha),\sin(\alpha),0)$.
Is this correct ?