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Let $f(\theta,\phi)=\frac{1}{\sqrt{2}}(\cos \theta,\sin \theta,\cos \phi,\sin \phi)$ be immersion of torus into $\mathbb R^4$. How to prove that $\nabla_{\frac{\partial}{\partial \theta}} \frac{\partial}{\partial \theta}=0$? It should be something easy, but I think I am confused a bit. Thank you.

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What does $\nabla_{\frac{\partial}{\partial \theta}} \frac{\partial}{\partial \theta}$ mean? –  Stephen Dedalus Dec 24 '14 at 10:12

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