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.
Vector field on torus as a submanifold of $\mathbb R^4$
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differential-geometry
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0What does $\nabla_{\frac{\partial}{\partial \theta}} \frac{\partial}{\partial \theta}$ mean? – 2014-12-24