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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 ?

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    $\begingroup$ Looks right to me. Are you able to visualize parallel transport along the great circles of the two-sphere? This makes it somewhat intuitive. $\endgroup$ Jan 29, 2012 at 9:08
  • $\begingroup$ Thank you. The solution looked idd correct after drawing those vectors. $\endgroup$
    – Nadori
    Jan 29, 2012 at 10:03

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