Define the concepts of (smooth) manifold and manifold with boundary for subsets of $\mathbf{R}^{N}$.

Let $X \subset \mathbf{R}^{6}$ be the subset defined by the equations

$$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}=1, \quad x_{4}^{2}-x_{5}^{2}-x_{6}^{2}=-1 .$$

Prove that $X$ is a manifold of dimension four.

For $a>0$, let $B(a) \subset \mathbf{R}^{6}$ denote the spherical ball $x_{1}^{2}+\ldots+x_{6}^{2} \leqslant a$. Prove that $X \cap B(a)$ is empty if $a<2$, is a manifold diffeomorphic to $S^{2} \times S^{1}$ if $a=2$, and is a manifold with boundary if $a>2$, with each component of the boundary diffeomorphic to $S^{2} \times S^{1}$.

[You may quote without proof any general results from lectures that you may need.]