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Hi everyone I need your help:

On the $n$-sphere we consider two different points with $p \neq -q$ and orthogonal between them. We take the curve $\gamma(t)=\cos(t^2)p+\sin(t^2)q $.

The exercise asks to compute the generic parallel transport from $t=0$ to $t=2\pi$.

I think this is a tricky exercise in the sense that it should be a happy idea behind the problem. I hope you can help me.

Thank you for your time.

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    1. Your curve $\gamma$ does not lie on the sphere unless $p$ and $q$ are orthogonal. 2. Did you mean "$\cos t$" instead of "$\cos(t^{2})$" (and similarly for $\sin t$)? 3. Did you really mean "parallel transport from $t = 0$ to $t = 2\pi$"? – 2017-01-18
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    Thank you for your observations i have edited the post. 3) Yes, from the point $\gamma (0)$ to $\gamma (2\pi)$ – 2017-01-18
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    $|\gamma|^2=\cos^4 t+\sin^4 t\lt 1$. – 2017-01-19
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    I edit the post Xipan, thanks you. – 2017-01-19

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