0
$\begingroup$

for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all?

Additionally, if we have a form a on manifold where it does not vanish, when we pull back the form by $\alpha^*$ which is corresponding to the coordinate patch at the point, what can we say about the k-form that we pulled back? Is it still nonvanishing?

$\endgroup$
  • 1
    $\begingroup$ It looks like the form vanishes on the line $\{(0,0,z)\}$, assuming the dimension is $3$ $\endgroup$ – Rolf Hoyer Apr 23 '15 at 20:35
0
$\begingroup$

It really depends on what you mean by the vanishing of $\omega$ at a point $p$. Do you mean that $\omega(p)(v,w) = 0$ for all $v,w\in\Bbb R^3$, or do you wish to find a submanifold $\alpha\colon M\hookrightarrow \Bbb R^3$ with the property that $\alpha^*\omega = 0$ (i.e., $\omega(p)(v,w)=0$ for all $p\in M$ and $v,w\in T_pM$)? For example, consider the surface $M=\{x^2-2yz=\text{constant}\}$. Then $\alpha^*\omega = 0$ on $M$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.