Consider a 2-surface $S$ defined in $\Bbb{R}^n$ ($n>3$) either directly as $F(x_1, x_2,... \, x_n) = 0$ or parametrically as $(x_1,... \, x_n) = (F_1(u, v),... F_n(u, v))$. Assume $S$ is simple connected, non singular, not self intersecting, and at least twice differentiable.
Consider $V$ is an $\Bbb{R}^3$ sub-manifold with coordinates linear in the $\Bbb{R}^n$. In other words, V is defined by rotation, translation, and dropping of $n-3$ coordinates. The objective for $V$ Is to enclose $S$. This may or may not be possible depending on $S$.
For example, a 2-sphere $x^2+y^2+z^2-R^2=0$ can be enclosed in $\Bbb{R}^3=(x, y, z)$ globally or locally, but a flat 2-torus $(x, y, z, w) = (R \cos(u), R \sin(u), P \cos(v), P \sin(v))$ cannot be enclosed in $\Bbb{R}^3$ (as defined above) either globally or even locally in a neighborhood of any point on the surface.
What necessary and sufficient conditions must $S$ satisfy for the existence of $V$ enclosing $S$ globally and/or locally?