Consider the following form on $\mathbb R^{2n + 2}$:
$$ \omega = \sum_{k=1}^{n+1}x_k dy_k – y_k dx_k$$
It defines a form on the sphere. Since recently I learned that in order to make it a form on the sphere I need to calculate $i^\ast \omega$ where $i: S^{2n +1} \hookrightarrow \mathbb R^{2n + 2}$ I tried to that. For the map $i$ I used
$(x_1, \dots, x_{n+1}, y_1, \dots, y_n, \sqrt{1-x_1^2 – \dots – y_n^2})\mapsto (x_1, \dots, x_{n+1}, y_1, \dots, y_n, \sqrt{1-x_1^2 – \dots – y_n^2})$
And found that the restriction is
$$ i^\ast \omega = \sum_{k=1}^n x_k dy_k – y_k dx_k + x_{n+1} \sum_{k=1}^{n+1} {-x_k\over \sqrt{1-x_1^2-\dots -y_n^2}} dx_k + x_{n+1}\sum_{k=1}^n {-y_k\over \sqrt{1-x_1^2-\dots -y_n^2}} dy_k – \sqrt{1-x_1^2-\dots -y_n^2} dx_{n+1}$$
Since I expected this to be something nice I believe what I did is not optimal or perhaps even wrong.
Now I am wondering:
To make a form on a manifold $M$ into a form on a submanifold $N$ is
it also possible to use charts for the pullback? Could I specify the
form just in such a local expression? And if so, is it equivalent to
expressing the pullback as a global expression?
Edit
Here's an example to make my question clearer:
An example on $S^1$ is $\omega = xdy – y dx$ (which is a form on $\mathbb R^2$). But $i^\ast \omega = d\theta$. That's the restriction calculated using the inclusion map. Alternatively, one can take a chart on $S^1$. E.g. projection onto $x$-axis. Then wlog $y=1$ (any non-zero constant will do) and so the resulting form is something like $\omega = -dx$. This seems to be equivalent to restriction by inclusion.
Best Answer
Assuming you want to use $(x_1,\dots,x_{n+1},y_1,\dots,y_n)$ as local coordinates on the sphere (where $y_{n+1}\ne 0$), here's a somewhat easier way to do your pullback calculation. Note that because $x_1^2+\dots+x_{n+1}^2+y_1^2+\dots+y_{n+1}^2=1$, we have $$\sum_{i=k}^{n+1} x_k\,dx_k + y_k\,dy_k = 0,$$ and so $$dy_{n+1} = -\frac1{y_{n+1}}\big(\sum_{k=1}^{n+1} x_k\,dx_k + \sum_{k=1}^n y_k\,dy_k\big).$$ Thus, the pullback of $\omega$ is $$\iota^*\omega = \sum_{k=1}^{n} \big(x_k-\frac{x_{n+1}y_k}{y_{n+1}}\big)dy_k -\sum_{k=1}^{n+1}\big(y_k-\frac{x_{n+1}x_k}{y_{n+1}}\big)dx_k.$$