Finding a chart around a point in a submanifold

Question

Define the set $M:= \{ x \in \mathbb R^4 : x_1x_4 - x_2x_3 =1 \}$, so $M$ is a 3-dimensional submanifold of $\mathbb R^4$.

I want to find a chart around $a=(a_1,a_2,a_3,a_4) \in M$, but I don't have an idea how to start constructing and would be glad for any help.

Thanks in advance!

Answer 1

Set $f := x_1x_4 - x_2x_3 - 1$. For all $x \in M$ we have $grad (f)(x) \neq 0$, hence M is a manifold. Consider a point $a=(a_1,a_2,a_3,a_4) \in M$, without loss of generality $a_1 \neq 0$. You get coordinate functions $$(x_1, x_2, x_3)$$

defined on the open subset $U_1:=\{x \in M: x_1 \neq 0\} \subset M$.

Note. On $U_1$ one has $x_4 = \frac {1+x_2 x_3}{x_1}$.