Submersions, level sets

[kellertuer]

It might sometimes be useful for embedded manifolds to work in the embedding and consider the level set function $h(\iota(p)) = 0$, where $\iota(p)$ is the embed function. This is closely related to submersions.
For both the derivative would be useful, for example to constrain $\text{Ker} Dh(p) = 0$ in the embedding, when trying to find a tangent vector (using constraints).

What would be the best way to name these (cf. a first idea also at #734)

• $h$ could be level_set(M,q) for any q from the embedding of M?
• $Dh$ would need a nice name
• submersions would probably need two manifolds (Domain and range)?

[kellertuer]

Revisiting this now, I would not be sure how most of our functions would work in this case (exp, log), so this is maybe something for another package with a slightly different focus even?

[mateuszbaran]

exp and log are difficult in general but there are still a few functions that I think fit decently well here, for example the Newton retraction: https://arxiv.org/abs/2006.14751 , and gradient and Hessian projections (in the Nicolas Boumal's book). Maybe there is also a cheap enough approximation to the vector transport.