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submanifold (Definition)

There are several conflicting definitions of what a submanifold is, depending on which author you are reading. All that agrees is that a submanifold is a subset of a manifold which is itself a manifold, however how structure is inherited from the ambient space is not generally agreed upon. So let's start with differentiable submanifolds of $ {\mathbb{R}}^n$ as that's the most useful case.

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 $ m$ continuously differentiable functions $ \rho_k \colon U \to {\mathbb{R}}$ where the differentials of $ \rho_k$ are linearly independent, such that
$\displaystyle M \cap U = \{ x \in U \mid \rho_k(x) = 0 , 1 \leq k \leq m \} .$    

Then $ M$ is called a submanifold of $ {\mathbb{R}}^n$ of dimension $ m$ and of codimension $ n-m$.

If $ \rho_k$ are in fact smooth then $ M$ is a smooth submanifold and similarly if $ \rho$ is real analytic then $ M$ is a real analytic submanifold. If we identify $ {\mathbb{R}}^{2n}$ with $ {\mathbb{C}}^n$ and we have a submanifold there it is called a real submanifold in $ {\mathbb{C}}^n$. $ \rho_k$ are usually called the local defining functions.

Let's now look at a more general definition. Let $ M$ be a manifold of dimension $ m$. A subset $ N \subset M$ is said to have the submanifold property if there exists an integer $ n \leq m$, such that for each $ p \in N$ there is a coordinate neighbourhood $ U$ and a coordinate function $ \varphi \colon U \to {\mathbb{R}}^m$ of $ M$ such that $ \varphi(p) = (0,0,0,\ldots,0)$, $ \varphi(U \cap N) = \{ x \in \varphi(U) \mid x_{n+1} = x_{n+2} = \ldots = x_m = 0 \}$ if $ n < m$ or $ N \cap U = U$ if $ n=m$.

Definition 2   Let $ M$ be a manifold of dimension $ m$. A subset $ N \subset M$ with the submanifold property for some $ n \leq m$ is called a submanifold of $ M$ of dimension $ n$ and of codimension $ m-n$.

The ambiguity arises about what topology we require $ N$ to have. Some authors require $ N$ to have the relative topology inherited from $ M$, others don't.

One could also mean that a subset is a submanifold if it is a disjoint union of submanifolds of different dimensions. It is not hard to see that if $ N$ is connected this is not an issue (whatever the topology on $ N$ is).

In case of differentiable manifolds, if we take $ N$ to be a subspace of $ M$ (the topology on $ N$ is the relative topology inherited from $ M$) and the differentiable structure of $ N$ to be the one determined by the coordinate neighbourhoods above then we call $ N$ a regular submanifold.

If $ N$ is a submanifold and the inclusion map $ i \colon N \to M$ is an imbedding, then we say that $ N$ is an imbedded (or embedded) submanifold of $ M$.

Definition 3   Let $ p \in M$ where $ M$ is a manifold. Then the equivalence class of all submanifolds $ N \subset M$ such that $ p \in N$ where we say $ N_1$ is equivalent to $ N_2$ if there is some open neighbourhood $ U$ of $ p$ such that $ N_1 \cap U = N_2 \cap U$ is called the germ of a submanifold through the point $ p$.

If $ N \subset M$ is an open subset of $ M$, then $ N$ is called the open submanifold of $ M$. This is the easiest class of examples of submanifolds.

Example of a submanifold (a regular and smooth submanifold in fact) is the unit sphere in $ {\mathbb{R}}^n$. This is in fact a hypersurface as it is of codimension 1.

Bibliography

1
William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
2
M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.



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

Also defines:  real submanifold, codimension of a manifold, local defining functions, real submanifold, smooth submanifold, real analytic submanifold, regular submanifold, imbedded submanifold, embedded submanifold, germ of a submanifold, open submanifold
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Cross-references: hypersurface, unit sphere, class, open subset, open, equivalent, equivalence class, imbedding, inclusion map, subspace, connected, disjoint union, mean, relative topology, topology, coordinate, integer, property, real analytic, smooth, codimension, dimension, linearly independent, functions, continuously differentiable, neighbourhood, point, differentiable, structure, manifold, subset, definitions
There are 40 references to this entry.

This is version 5 of submanifold, born on 2004-11-02, modified 2007-12-18.
Object id is 6440, canonical name is Submanifold.
Accessed 11013 times total.

Classification:
AMS MSC: 57N99 (Manifolds and cell complexes :: Topological manifolds :: Miscellaneous)
 53B25 (Differential geometry :: Local differential geometry :: Local submanifolds)
 53C40 (Differential geometry :: Global differential geometry :: Global submanifolds)
 32V40 (Several complex variables and analytic spaces :: CR manifolds :: Real submanifolds in complex manifolds)
Pending Errata and Addenda
None.
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