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I have a smooth diffeomorphism $\Phi\colon M \to N$ between two 2D hypersurfaces in $\mathbb{R}^3$, eg. the sphere and some deformed version of the sphere.

If I have a unit (outward) normal vector $n$ at some point $p \in M$, is there a formula to get the unit (outward) normal $\tilde n$ at the point $\Phi(p) \in N$ in terms of $n$?

Maybe (or not) something like $\tilde n = M\Phi(n)$ (where $\Phi$ is applied to each component on $n$, and $M$ is some matrix)?

I need this because I wish to describe the normal field of $N$, which is a complicated surface, using the normal field on $M$ (which is the sphere in my example).

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  • $\begingroup$ Since $\Phi$ is just defined on $M$, you can't consider something like $\Phi(n)$ (which presumably means $d_p\Phi(n_p)$?) $\endgroup$
    – Didier
    Commented Mar 26, 2022 at 16:05
  • $\begingroup$ @Didier Yes, you're right. I don't know, there be may some other related objects. I just want a formula for $\tilde n$ in terms of $n$. $\endgroup$
    – BBB
    Commented Mar 26, 2022 at 16:29
  • $\begingroup$ Unless you have $\Phi$ as a mapping to $\Bbb R^3$, this seems totally impossible. The diffeo is otherwise intrinsic, basically (locallyj a diffeomorphism between open subsets of $\Bbb R^2$. $\endgroup$
    – Ted Shifrin
    Commented Mar 26, 2022 at 17:01

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