Oriented Atlas for $\mathbb{RP}^3$

Question

I know $\mathbb{RP}^3$ is orientable, but it came as a surprise that I don't seem to find an oriented atlas for it so easily:

If I set the standard atlas

$(x_1, x_2, x_3)\mapsto [x_1:x_2:x_3:1]$

$(y_1, y_2, y_3)\mapsto [y_1: y_2: 1: y_3]$

$(z_1, z_2, z_3)\mapsto [z_1: 1: z_2: y_3]$

$(w_1, w_2, w_3)\mapsto [1: w_1: w_2: w_3]$

Then we may look at the transition from first to second:
$(x_1, x_2, x_3)\mapsto \left( \dfrac{x_1}{x_3}, \dfrac{x_2}{x_3}, \dfrac{1}{x_3} \right).$
The Jacobian has determinant $-\dfrac{1}{x_3^4}< 0$.

So we can fix this by changing the second chart to $(y_1, y_2, y_3)\mapsto [y_1: y_2: -1: y_3]$, for instance.

The same problem occurs for the other two so I can change my atlas to

$(x_1, x_2, x_3)\mapsto [x_1:x_2:x_3:1]$

$(y_1, y_2, y_3)\mapsto [y_1: y_2: -1: y_3]$

$(z_1, z_2, z_3)\mapsto [z_1: -1: z_2: y_3]$

$(w_1, w_2, w_3)\mapsto [-1: w_1: w_2: w_3]$

in the hopes that now it will be fixed. But then it's not hard to see the Jacobian of the second with the third chart will have negative Jacobian. I could fix it by changing one sign, but then it will generate a negative Jacobian again on the transition with the first. In other words, it seems that there's no immediate way to fix this.

One interesting thing is that with the circle this works seamlessly, because there are only two charts this way.

On the other hand, my intuition would make me think that because this standard chart will definitely be diffeomorphic to whatever oriented atlas I can find (though not orientation-preserving diffeomorphic), I should be able to make this work by just changing signs.

Would anyone have some insight on finding an oriented atlas?

Answer 1

The same issue occurs all over with having to choose "arbitrary" orientations when you omit one variable. The negative sign in the second component of the cross product (or the correspondence $(a,b,c)\leftrightsquigarrow a\,dy\wedge dz + b\,dz\wedge dx + c\, dx\wedge dy$ with $dz\wedge dx$ rather than $dx\wedge dz$) is a manifestation of the same phenomenon. So the clue is to permute the variables.

Using $[x_1:x_2:x_3:1]$, now try $[y_2:y_1:1:y_3]$, $[z_1:1:z_2:z_3]$, and $[1:w_1:w_3:w_2]$.