Solution
= Solution
Let
$$
F(x,y,z)=x^2+y^2+z^2-2xy-2yz.
$$
Its <gradient> is
$$
\nabla F=2(x-y,\ y-x-z,\ z-y).
$$
If this vanishes, the first and third coordinates give $x=y=z$, and the middle coordinate then gives $y=0$. Thus the origin is the only <critical point> of $F$. It has been removed from $V$, so $0$ is a <regular value> of the restriction under consideration. The <regular level set theorem> shows that
$$
V=F^{-1}(0)\setminus\{0\}
$$
is a smooth surface.