Give a volume form on $\mathbb{RP}^3$

Question

I was asked to determine a volume form on $\mathbb{RP}^3$. I would really appreciate any help. Thanks in advance.

Answer 1

There is a volume form $\omega$ on $S^3$ given by the pullback by the inclusion map of the form

$$x_1 dx_2 \wedge dx_3 \wedge dx_4 - x_2 dx_2 \wedge dx_3 \wedge dx_4 + x_3 dx_1 \wedge dx_2 \wedge dx_4 - x_4 dx_1 \wedge dx_2 \wedge dx_3$$

on $\mathbb{R}^4$. Since $3$ is odd, you can check that this form is invariant under the antipodal map $A:S^3 \to S^3$, $A(v) = -v$, i.e., that $A^*\omega = \omega$. (More generally, for $S^n$ inside $\mathbb{R}^{n+1}$, we have $A^*\omega = (-1)^{n+1}\omega$.)

Thus the quotient map $S^3 \to \mathbb{RP}^3$, which is a local diffeomorphism, lets us push forward $\omega$ to a well-defined volume form on $\mathbb{RP}^3$.