Is the solution space to $x_1x_4-x_2x_3=0$ in $\mathbb{R}$ a manifold? I was wondering whether the set of $(x_1,x_2,x_3,x_4)$ that solve the equation $x_1x_4-x_2x_3=0$ in $\mathbb{R}$ is a manifold. My first guess is that it is most likely not because if I define a function $J(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$ then $0$ is not a regular value, as $dJ(0,0,0,0)=0.$ But this does afaik not mean that it cannot be a manifold.
If anything is unclear, please let me know.

No, it is not a manifold because it is a cone with apex the origin $O$, and a cone that is not a vector subspace  is never a manifold.
The fact that $dJ(0,0,0,0)=0$ indeed essentially  implies that $J=0$ is not a manifold at $O$, but this is a very subtle point never explained in differential geometry books.
 See here for more precise explanations .
