2
$\begingroup$

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?

  • 0
    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

1 Answers 1

1

$(i^{*}w)(p)v=w(i(p))(di_{p}v)=dz(di_{p}v)=dz(i(v))=dz(v)=0 \Leftrightarrow v=(v_{1}, v_{2},0) $, where $p\in \mathbb{S} ^{2}$,and$ v\in \mathbb{R} ^{3}$.

  • 0
    @Gobi, My answer did not help? You do not agree?2013-02-10