If is in fact smooth then is a smooth hypersurface and similarly if is real analytic then is a real analytic hypersurface. If we identify with and we have a hypersurface there it is called a real hypersurface in . is usually called the local defining function. Hypersurface is really special name for a submanifold of codimension 1. In fact if 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 a singular hypersurface. That is the definition is the same as above, but we do not require . 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 with smooth boundary.
An example of a singular hypersurface in is for example the zero set of 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.
|Date of creation||2013-03-22 14:32:56|
|Last modified on||2013-03-22 14:32:56|
|Last modified by||jirka (4157)|
|Defines||real analytic hypersurface|
|Defines||local defining function|