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Let $i:S^2 \to \mathbb{R}^3$ be inclusion. Let $\omega=dz$ be a 1-form on $\mathbb{R}^3$. I want to compute $i^* \omega$ and it vainishes exactly two points.

Is it right that $i^* \omega=i^*dz=d(i^*z)=d(z \circ i)$? Why does it vanishes at two points?

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    Hint: $dz$ is the differential of projection to the $z$-axis. For which two points of $\mathbb{S}^2$ does the projection to the $z$ axis not have full rank?2012-12-12

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