Let M be a subset of n such that for every point pM there exists a neighbourhood Up of p in n and a continuously differentiable function ρ:U with gradρ0 on U, such that


Then M is called a hypersurface.

If ρ is in fact smooth then M is a smooth hypersurface and similarly if ρ is real analytic then M is a real analytic hypersurface. If we identify 2n with n and we have a hypersurface there it is called a real hypersurface in n. ρ is usually called the local defining function. Hypersurface is really special name for a submanifoldMathworldPlanetmath of codimension 1. In fact if M is just a topological manifoldMathworldPlanetmathPlanetmath 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 setPlanetmathPlanetmath 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 gradρ0. 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 n with smooth boundary.

An example of a singular hypersurface in 2 is for example the zero set of ρ(x1,x2)=x1x2 which is really just the two axis. Note that this hypersurface fails to be a manifold at the origin.


  • 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
Title hypersurface
Canonical name Hypersurface
Date of creation 2013-03-22 14:32:56
Last modified on 2013-03-22 14:32:56
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 32V40
Classification msc 14J70
Related topic Submanifold
Defines smooth hypersurface
Defines real analytic hypersurface
Defines real hypersurface
Defines local defining function
Defines singular hypersurface
Defines non-singular hypersurface
Defines hypervariety