Why this manifold can not be embbeded in 3d Euclidean space?

Question

Consider the following system of 6 equations in 9 variables $x_1, x_2,x_3,...x_9$

$ x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2 + = 1 $

$ x_4 ^ 2 + x_5 ^ 2 + x_6 ^ 2 + = 1 $

$ x_7 ^ 2 + x_8 ^ 2 + x_9 ^ 2 + = 1 $

$ x_1 x_4 + x_2 x_6 + x_3 x_7 = 0 $

$ x_1 x_7 + x_2 x_8 + x_3 x_9 = 0 $

$ x_4 x_7 + x_5 x_8 + x_6 x_9 = 0 $

The solution space of this system is given as an example of a three dimensional compact manifold which can not be fully embedded in 3D but at any point, for instance $\left(1,0,0,0,1,0,0,0,1,\right)$, one can find a neighborhood of it which is in one to one correspondence with some neighborhood of Euclidean space.

Thank you

Answer 1

No compact $n$-manifold without boundary can be embedded in $\Bbb R^n$. -- Mike Miller

Indeed, since any $n$-manifold without boundary is locally homeomorphic to $\mathbb{R}^n$, the invariance of domain implies that its image under an embedding into $\mathbb{R}^n$ must be an open set. In particular, it cannot be compact.