How do I prove that a set of equations define a submanifold?

Question

I have two equations of 4 variables, and want to prove that they define a submanifold.

I showed that two of the variables are defined by the other two, and wanted to get a graph out of it to show that it is a submanifold. However, I have two problems:

1. I have the equations simplified into $x_2, x_3$ defined by $x_1, x_4$. So if I find a smooth function mapping $(x_1, x_4)$ to $(x_2, x_3)$, I would have the graph as $(x_1, x_4, x_2, x_3)$, and that it is a submanifold. How do I get from there to that the set $(x_1, x_2, x_3, x_4)$ is a submanifold?

2. I got $x_3^2$ is defined by $x_1, x_4$ ($x_3^2=1-x_1^4-x_4^2$) but couldn't simplify it further. How do I get a smooth function mapping $x_1, x_4$ to $x_3$ from that?

Sorry for the lack of context--I don't want to get too specific to avoid cheating, but I can clarify further if needed.

Answer 1

1. You are getting hung up on an artificial ordering. $(x_1, x_2, x_3, x_4)$ also counts as a graph of your map.
2. if $1-x_1^4 -x_4^2$ is ever $0$, then you've chosen the wrong domain for this particular map. You will need to figure out a different mapping around those points. If you've restricted your mapping to a domain where this is never $0$, then you choose sign for the square root. This choice defines a smooth mapping that covers a part of the submanifold. The other choice of sign defines a different smooth mapping that covers a different part of the submanifold. You cannot choose a mapping that works everywhere, but this is normal for manifolds. It is why the concept of manifold was invented in the first place.