If $\rho$ is in fact smooth then $M$ is a smooth hypersurface and similarly if $\rho$ is real analytic then $M$ is a real analytic hypersurface. If we identify ${\mathbb{R}}^{{2n}}$ with ${\mathbb{C}}^{n}$ and we have a hypersurface there it is called a real hypersurface in ${\mathbb{C}}^{n}$. $\rho$ is usually called the local defining function. Hypersurface is really special name for a submanifold of codimension 1. In fact if $M$ is just a topological manifold of codimension 1, then it is often also called a hypersurface.

A real or complex analytic subvariety of codimension 1 (the zero set of a real or complex analytic function) is called a singular hypersurface. That is the definition is the same as above, but we do not require $\operatorname{grad}\rho\not=0$. Note that some authors leave out the word singular and then use non-singular hypersurface for a hypersurface which is also a manifold. Some authors use the word hypervariety to describe a singular hypersurface.

An example of a hypersurface is the hypersphere (of radius 1 for simplicity) which has the defining equation

Another example of a hypersurface would be the boundary of a domain in ${\mathbb{C}}^{n}$ with smooth boundary.

An example of a singular hypersurface in ${\mathbb{R}}^{2}$ is for example the zero set of $\rho(x_{1},x_{2})=x_{1}x_{2}$ which is really just the two axis. Note that this hypersurface fails to be a manifold at the origin.