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hypersurface (Definition)
Definition 1   Let $ M$ be a subset of $ {\mathbb{R}}^n$ such that for every point $ p \in M$ there exists a neighbourhood $ U_p$ of $ p$ in $ {\mathbb{R}}^n$ and a continuously differentiable function $ \rho \colon U \to {\mathbb{R}}$ with $ \operatorname{grad} \rho \not= 0$ on $ U$, such that
$\displaystyle M \cap U = \{ x \in U \mid \rho(x) = 0 \} .$    

Then $ M$ is called a hypersurface.

If $ \rho$ is in fact smooth then $ M$ is a smooth hypersurface 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

$\displaystyle x_1^2 + x_2^2 + \ldots + x_n^2 = 1 .$    

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.



"hypersurface" is owned by jirka.
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See Also: submanifold

Also defines:  smooth hypersurface, real analytic hypersurface, real hypersurface, local defining function, singular hypersurface, non-singular hypersurface, hypervariety

Attachments:
Lewy hypersurface (Example) by jirka
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Cross-references: origin, axis, domain, boundary, equation, radius, hypersphere, manifold, singular, real, zero set, complex analytic subvariety, topological manifold, codimension, submanifold, real analytic, smooth, function, continuously differentiable, neighbourhood, point, subset
There are 11 references to this entry.

This is version 5 of hypersurface, born on 2004-08-11, modified 2007-12-18.
Object id is 6099, canonical name is Hypersurface.
Accessed 7799 times total.

Classification:
AMS MSC: 14J70 (Algebraic geometry :: Surfaces and higher-dimensional varieties :: Hypersurfaces)
 32V40 (Several complex variables and analytic spaces :: CR manifolds :: Real submanifolds in complex manifolds)
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