# Show that the standard atlas for $\mathbb{RP}^2$, ${(U_i,\phi_i)}_{i=1}^3$, is not orienting.

Show that the standard atlas for $$\mathbb{RP}^2$$, $${(U_i,\phi_i)}_{i=1}^3$$, is not orienting.

$$U_1=\{[x^1,x^2,x^3]\in \mathbb{RP}^2 \ | \ x_1 \neq 0\}$$, $$\phi_1 ([x^1,x^2,x^3])=(\frac{x^2}{x^1},\frac{x^3}{x^1})$$

My attempt:

An atlas $$\mathcal{A}$$ is orienting if any two charts in $$\mathcal{A}$$ are orientation compatible. It suffices to prove that there are 2 charts that are not orientation compatible in the standard atlas for $$\mathbb{RP}^2$$.

Let's check the compatibility of the charts $$(U_1,\phi_1)$$ and $$(U_3,\phi_3)$$

$$\phi_3 \circ (\phi_1)^{-1} (x^1,x^2)=\phi_3 ([1,x^1,x^2])=(\frac{1}{x^2},\frac{x^1}{x^2})$$

$$[(\phi_3 \circ (\phi_1 )^{-1})_*]= \begin{bmatrix} 0 & \frac{-1}{(x^2)^2} \\ \frac{1}{x^2} & \frac{-x^1}{(x^2)^2} \end{bmatrix}$$

$$\text{det} [(\phi_3 \circ (\phi_1 )^{-1})_*] =\frac{1}{(x^2)^3}$$

I could not decide weather this is positive or negative in $$\phi_1 (U_1 \cap U_3)$$. I know that

$$U_1 \cap U_3=\{[x^1,x^2,x^3] \ | \ x^1,x^3\neq 0 \}$$

$$\phi_1 (U_1 \cap U_3)=\{(\frac{x^2}{x^1},\frac{x^3}{x^1}) \ | \ x^1,x^3\neq 0 \}=\mathbb{R}^2 \setminus \{y=0\}$$

But still cannot figure it out.

I also calculated $$\text{det} [(\phi_2 \circ (\phi_1 )^{-1})_*] =\frac{-1}{(x^1)^3}$$, and $$\text{det} [(\phi_3 \circ (\phi_2 )^{-1})_*] =\frac{-1}{(x^2)^3}$$.

I still do not know weather they're positive or negative. Any help would be appreciated.

$$\frac{1}{x^3}$$ has the same sign as $$x$$. $$U_1 \cap U_3$$ has points with positive and negative $$x_2$$, and so positive and negative determinant, so it’s not orientation preserving.