Exercise: Compute a distribution given the integral manifold

Question

I am new in Stack Mathematics. I need your help in solving the follow exercise.

"Compute a distribution $\Delta$ over $\mathbb{R}^3$ whose integral manifold is the surface of a sphere (i.e. the set of points at the same distance $r$ from the center)."

Usually, the distribution is given and I have to find, if it exists, the integral manifold, but this time is quite the opposite. Does anyone have any idea how to solve this kind of exercises?

Thank you in advance!

Answer 1

Hint Without loss of generality take the center to be the origin ${\bf 0}$, and consider any point $p := (x, y, z) \neq {\bf 0}$. Since the sphere $S_{|p|}^2$ of radius $|p|$ centered at $\bf 0$ is an integral manifold of the distribution $D$, $$D_p = T_p S_{|p|}^2 .$$

Additional hint With respect to the Euclidean metric (and invoking the canonical identification of the tangent space to a vector space at a point with the vector space itself), $$T_p S_{|p|}^2 = \{p\}^{\perp} .$$