Proving Manifold Problems in R^4: X at a Point a

['AQF]

We have a subset X, which is contained in R^4 (i.e., it is contained in the reals in 4 dimensions).
(a) We must prove that the following two equations represent a manifold in the neighborhood of the point a = (1,0,1,0):
(x_1)^2+(x_2)^2-(x_3)^2-(x_4)^2=0 and x_1+2x_2+3x_3+4x_4=4.
(b) Also we must find a tangent space to X at a.
(c) We must find a pair of variables that the equations above do not express as functions of the other two.
(d) We must determine whether the enter set X is a manifold and prove the conclusion.

How do you do this problem?
Thanks.

 

[hypermorphism]

Use the definition of a manifold (or a theorem thereof). For example, a theorem dealing with level sets.

 

['AQF]

Can you elaborate, especially on level sets?

 

[hypermorphism]

If g:Rⁿ -> R^m, you may have a theorem that states the conditions necessary for g^-1(0) to be an (n-m)-dim. manifold. Such a set is called a level set of g.
For example, if g:R³ -> R is the function g(x) = ||x|| - 1, then g^-1(0) is the 2-sphere.
The conditions in my book are that the domain of g be open, and that g be differentiable with rank m wherever g(x)=0.