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 \begin{equation*} x_1^2 + x_2^2 + \ldots + x_n^2 = 1 . \end{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.

Bibliography

1

M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.