= Hypersurface
{wiki}
A smooth <hypersurface> in an $n$-dimensional <smooth manifold> is an <embedded submanifold> of codimension one. Locally it can be written as $\{\phi=0\}$ with
$$
d\phi_p\ne0
$$
at every <point> of that zero set. The <regular level set theorem> gives this local description, and its <tangent space> is $\ker d\phi_p$.
The <covector> $d\phi$ is defined without a metric. If the ambient manifold has a <Riemannian metric>, its metric dual <gradient> $\nabla\phi$ is a <normal vector>, since $g(\nabla\phi,v)=d\phi(v)=0$ for every tangent $v$. A <pseudo-Riemannian metric> gives the analogous normal, which can be null. A global scalar defining <function> need not exist, as the <normal-bundle obstruction to being a regular level set> shows.
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