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$\gamma (s)$ is unit speed curve on a surface M in $R^3$, and S is its intrinsic normal. How can i prove that S is parallel along $\gamma$ if and only if $\gamma$ is geodesic?

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Writing $\gamma' = T$, recall that $S=n\times T$, where $n$ is the surface normal. To say that $S$ is parallel along $\gamma$ is to say that $0=\nabla_T S = (S'\cdot T)T$, since $S'\cdot S = 0$. But $S'\cdot T = -T'\cdot S$ since $S\cdot T = 0$. And $T'\cdot S = 0$, i.e., $T'$ is normal to the surface, if and only if $\nabla_T T = 0$ if and only if $\gamma$ is a geodesic. So $S$ is parallel along $\gamma$ if and only if $\gamma$ is a geodesic.

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