$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Question

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$.

Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$.

Check if the obtained $1$-form respects or not the induced orientation.

Honestly I don't know where to begin, some help would be appreciated.

Somehow the answer for the first part is $\frac{e^{xy}}{y}dx$ or $-\frac{e^{xy}}{x}dy$.

The induced orientation of $S^1$ is given by $dx$ (this I know).

Thanks in advance.

Answer 1

Assuming Euclidean metric on $\mathbb R^2$ and given a weighted "volume form" $\omega=e^{xy} dx \wedge dy$ which is not the natural volume form for this metric, we're asked to find its induced "volume form" $i^{*}(\iota_\nu \omega)$ on $S^1$, where we first take the interior product with the unit outer normal $\nu=x\partial_x+y\partial_y$ and then pull back to $S^1$ along the inclusion map.

Using the linearity of the interior product we arrive directly at @James's result: $\iota_\nu \omega=(-ydx+xdy)e^{xy}$. One could transform it further to either $\frac{e^{xy}}{y}dx$ or $-\frac{e^{xy}}{x}dy$ by differentiating and using again the $x^2+y^2=1$ and then for some reason changing the sign as @Ted noted.

Answer 2

I guess what I write here is wrong. But, I'm not sure exactly why, so I post it here in the hope someone points out the error of my ways. The two-form $\omega = e^{xy}dx \wedge dy$ may be written as $\omega = e^{r^2 \sin \theta \cos \theta} r \, dr \wedge d\theta$. Then, in polar coordinates, the unit circle is $r=1$ and $\partial / \partial r$ is an outward-pointing vector field on the circle. If I understand the induced orientation of $\omega$ correctly, then all I need to do is to take the inner product (sometimes called "hook") of $\omega$ with $\partial/\partial r$ and pull that back to the circle. Calculate $$ \omega(\partial/\partial r, \cdot) = e^{r^2 \sin \theta \cos \theta} r dr(\partial/r)d\theta = e^{r^2 \sin \theta \cos \theta} r d\theta$$ but, this is to be pulled-back to the circle. In Cartesians, $$ d\theta = \frac{-ydx+xdy}{x^2+y^2} $$ thus, using a hybrid notation momentarily, $$ \omega(\partial/\partial r, \cdot) = e^{r^2 \sin \theta \cos \theta} r \frac{-ydx+xdy}{x^2+y^2} = \frac{-ydx+xdy}{\sqrt{x^2+y^2}}e^{xy}$$ But, I would think the pull-back of this form to the curve $x^2+y^2=1$ simply gives us: $$ \phi = (-ydx+xdy)e^{xy}$$ Given all this, I'd think $\phi$ is consistent with $dx$ when $y<0$ but, opposite $dx$ when $y>0$. I'm not sure how to get the quoted answers in the stated question. Roughly I follow Lee's Smooth Manifolds, which is also discussed in Inducing orientations on boundary manifolds .