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I have an m-dimensional riemannian manifold M and an n-dimensional submanifold N that is given by $N = f^{-1}(0)$, where $f: M \longrightarrow \mathbb{R}^{m-n}$ ($0$ is supposed to be a regular value of f).

How can I express the second fundamental form of N in terms of f?

\Edit: I forgot to clarify what $f$ is

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    Let's start with the dimension of $N$. It will be $m-1$, that is with a single defining function you get a hypersurface. Do you know the expression for the second fundamental form in this case?2012-10-25
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    No, as a matter of fact, I do not :-/2012-10-25

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