Suppose I have a smooth function $u:\mathbb R^n\to\mathbb R$. Assume that its gradient doesn't vanish (near any point where we investigate it). Is there a list of different (intrinsic and extrinsic) curvature quantities of level sets of $u$ in terms of derivatives of $u$? I have been unable to find such a list.
The level set is a Riemannian manifold and its curvature can be described by various curvature tensors. It is also a submanifold of the ambient $\mathbb R^n$ and the second fundamental form describes its curvature as a submanifold. These are what I refer to as intrinsic and extrinsic curvature quantities.
Here are two examples of questions that the list should answer. I am looking for a resource that would contain the answer to these two questions and many others, not just the answer to these two. These example questions give a criterion for what I am looking for, that's all. This question is a reference request.
- If $n=3$, what is the Gaussian curvature of $u^{-1}(u(0))$ at $0$ in terms of derivatives of $u$?
- How to express the mean curvature of the level set in terms of derivatives of $u$ in any dimension?
For the first, you might want to look at a tech report I wrote several years back:
http://cs.brown.edu/people/jhughes/papers/Hughes-DGO-2003/paper.pdf
For the second, there's a quite general paper (of which why tech report is a distillation in the 3D case, with a correction):
Peter Dombrowski. Krümmungsgrößen gleichungsdefinierter untermannigfaltigkeiten riemmannscher mannigfaltigkeiten. Mathematische Nachrichten, 38(3/4):133–180, 1968.