Checking if subset of $\mathbb{R}^3$ is $C^1$-manifold

Question

I want to check if

$M := \{x \in \mathbb{R}^3 | x_1 + x_2 +x_3 =0, x_1 ^2 + x_2 ^2 + x_3 ^2 =1 \} $

is a $C^1$-manifold.

To do this I would need to find for every point $x \in M$ a $C^1$-function $\phi : U \rightarrow ?$ with the known properties, but I do not have an idea for such a function, and if there should not exist one, I also do not see an argument which would show this.

Answer 1

Hint: Consider $f:R^3\rightarrow R^2$ defined by $f(x_1,x_2,x_3)=(x^2_1+x^2_2+x^2_3-1,x_1+x_2+x_3)$, $M=f^{-1}(0)$, compute the Jacobian of $f$ and see if its rank is 2 on $M$.

$Jac(f)=\pmatrix{2x_1 & 2x_2 & 2x_3\cr 1 &1 &1 }$ The minors are $2(x_1-x_2), 2(x_1-x_3), 2(x_2-x_3)$ vanish if $x_1=x_2=x_3$, but if $(x,x,x)\in M, 3x=0$ so $x=0$ impossible since we must have $3x^2=1$. Thus the restriction of $f$ on a neighborhood of $M$ is a submersion and $M$ is a submanifold.