Suppose $\omega$ is a $n-1$ form on a $n-1$ dimensional manifold and $(a_1(x)dx_1 + ... + a_n(x)dx_n )\wedge \omega = c\Omega $, with $c \neq 0$ and $\Omega =dx_1\wedge...\wedge dx_n$. Moreover $(a_1(x),...,a_n(x))= a(x)$ is a normal vector on the manifold, with a nonzero norm. Apparently with this you can say there exists a normal unit vector field on $M$, where you say $n(x) = \frac{a(x)}{\| a(x)| }$ if $c >0$ and $n(x) = -\frac{a(x)}{\| a(x)| }$ if $c < 0$. Why can you say this, how is from the wedge product seen that the normal vector fields is smooth, as in it "doesn't keep switching" .

  • $\begingroup$ Is the manifold embedded in $\Bbb R^n$? $\endgroup$ – edm Dec 5 '18 at 4:03
  • $\begingroup$ Yeeah it is in Rn $\endgroup$ – AkatsukiMaliki Dec 5 '18 at 7:06
  • $\begingroup$ And is $a_1(x)dx_1 + \cdots + a_n(x)dx_n$ defined on the manifold or on the whole $\Bbb R^n$? $\endgroup$ – edm Dec 5 '18 at 11:07

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