# Are principal curvatures equal?

1. Jul 10, 2012

### Thinkor

This is a question about Newtonian gravity. I post it here, because there seems little interest in gravity under the classical physics section.

Principal curvatures on surfaces of equal potential around an isolated spherically symmetric orb are equal at every point by symmetry, but are they equal generally in Newtonian gravity? If not, do you have an example where they would not be equal? If so, do you have a reference that proves it?

2. Jul 10, 2012

### Thinkor

Actually, immediately after posting this I thought of a counterexample. Imagine two flat walls in space close to one another but with a gap between them. The field is flat between them if sufficiently far from the edges of the walls. Now, crease both walls equally along the same line. The principal curvature along the midline between the creased walls will be zero (if far enough from the edges), but the other principal curvature for points on the midline will be nonzero.