We have the volume form $\mu=i^*(\omega)$ on $S^n$, where $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n+1}$$ is a k-form on $\mathbb R^{n+1}$ and $i:S^n\to \mathbb R^{n+1}$ is the inclusion.

Define the antipodal map $A:S^n\to S^n$ by $x \mapsto -x$. Indeed, $A^*(\mu)=(-1)^{n+1}\mu$. Does this gives us a volume form on $\mathbb R P^n$ for n odd? It seems we have the projection map $p:S^n\to \mathbb R P^n$, but it's the wrong direction to pull back the form. Any help will be appreciated.

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    $\begingroup$ This may be of help. $\endgroup$ – Neal Dec 18 '16 at 2:26

HINT: Pull back by local inverses of $p$, and use your computation to check that you get a well-defined form on $\Bbb RP^n$. Alternatively, define a form $\eta$ on $\Bbb RP^n$ by setting its value $\eta([x])(v_1,\dots,v_n)$ to be something in terms of $\mu$, and check that everything is well-defined.


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