Defining hypersurfaces in terms of algebraic surfaces

Question

If in Euclidean space one has an equation of the form $f(p) = 0$, then the solutions $\lbrace p\in \mathbb{R}^{n}: f(p) = 0\rbrace$ determine a hypersurface in Euclidean space. Given a Lorentzian manifold, how does one generalise this in terms of charts on the manifold? More precisely, I would like to know if given a function $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, how one fixes charts on the manifold such that the zero set of a function as defined above in Euclidean space can be identified with a hypersurface in the manifold.

Answer 1

This construction is independent of the metric of the manifold. Precisely, if $f:M\to \Bbb R$ is a smooth function, with $M$ a smooth manifold, and if $a\in \Bbb R$ has the property that $\mathrm df_p:T_pM\to \Bbb R$ is surjective for every $p\in f^{-1}(a)$, then $N=f^{-1}(a)$ is a smooth hypersurface in $M$.