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# submanifold

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.

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

$M\cap U=\{x\in U\mid\rho_{k}(x)=0,1\leq k\leq m\}.$ |

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.

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.

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.

# References

- 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.

## Mathematics Subject Classification

32V40*no label found*53C40

*no label found*53B25

*no label found*57N99

*no label found*

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