hypersurface
Definition.
Let M be a subset of ℝn such that for every point p∈M there exists a neighbourhood Up of p in ℝn and a continuously differentiable function ρ:U→ℝ with gradρ≠0 on U, such that
M∩U={x∈U∣ρ(x)=0}. |
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 ℝ2n with ℂn and we have a
hypersurface there it is called a real hypersurface in
ℂn. ρ 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 a
singular hypersurface. That is the definition is the same as above, but
we do not require gradρ≠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
x21+x22+…+x2n=1. |
Another example of a hypersurface would be the boundary of a domain in ℂn with smooth boundary.
An example of a singular hypersurface in ℝ2 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.
References
- 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 |