hypersurface

Definition.

Let $M$ be a subset of ${ℝ}^n$ such that for every point$p\in M$ there exists a neighbourhood $U_p$ of $p$ in ${ℝ}^n$and a continuously differentiable function $\rho :U\to ℝ$ with$\mathrm{grad}\rho \ne 0$ on $U$,such that

$$M\cap U=\{x\in U\mid \rho (x)=0\}.$$

Then $M$ is called a hypersurface.

If $\rho$ is in fact smooth then $M$ is a smooth hypersurface andsimilarly if $\rho$ is real analytic then $M$ is a real analytichypersurface. Ifwe identify ${ℝ}^{2n}$ with ${ℂ}^n$ and we have ahypersurface there it is called a real hypersurface in${ℂ}^n$. $\rho$ 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 (http://planetmath.org/RealAnalyticSubvariety) or complex analytic subvariety of codimension 1 (the zero set of a real or complex analytic function) is called asingular hypersurface. That is the definition is the same as above, butwe do not require $\mathrm{grad}\rho \ne 0$. 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

$$x_1^2+x_2^2+\mathrm{\dots }+x_n^2=1.$$

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

An example of a singular hypersurface in ${ℝ}^2$ is for example the zero setof $\rho (x_1,x_2)=x_1{x}_2$ which is really just the two axis. Note that thishypersurface fails to be a manifold at the origin.