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 ${ℝ}^n$ as that’s the most useful case.

Definition.

Let $M$ be a subset of ${ℝ}^n$ such that for every point $p\in M$ there exists a neighbourhood $U_p$ of $p$ in ${ℝ}^n$ and $m$ continuously differentiable functions ${\rho }_k:U\to ℝ$ where the differentials of ${\rho }_k$ are linearly independent, such that

$$M\cap U=\{x\in U\mid {\rho }_k(x)=0,1\le k\le m\}.$$

Then $M$ is called a submanifold of ${ℝ}^n$ 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 ${ℝ}^{2n}$ with ${ℂ}^n$ and we have a submanifold there it is called a real submanifold in ${ℂ}^n$. ${\rho }_k$ 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 $n\le m$, such that for each $p\in N$ there is a coordinate neighbourhood $U$ and a coordinate function $\phi :U\to {ℝ}^m$ of $M$ such that $\phi (p)=(0,0,0,\mathrm{\dots },0)$, $\phi (U\cap N)=\{x\in \phi (U)\mid x_{n+1}=x_{n+2}=\mathrm{\dots }=x_m=0\}$ if 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 $n\le m$ 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 $i:N\to M$ 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 in fact) is the unit sphere in ${ℝ}^n$. This is in fact a hypersurface as it is of codimension 1.

Title	submanifold
Canonical name	Submanifold
Date of creation	2013-03-22 14:47:20
Last modified on	2013-03-22 14:47:20
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	8
Author	jirka (4157)
Entry type	Definition
Classification	msc 32V40
Classification	msc 53C40
Classification	msc 53B25
Classification	msc 57N99
Related topic	Manifold
Related topic	Hypersurface
Defines	real submanifold
Defines	codimension of a manifold
Defines	local defining functions
Defines	real submanifold
Defines	smooth submanifold
Defines	real analytic submanifold
Defines	regular submanifold
Defines	imbedded submanifold
Defines	embedded submanifold
Defines	germ of a submanifold
Defines	open submanifold