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 andsimilarly if $\rho$ is real analytic then $M$ is a real analyticsubmanifold. Ifwe identify ${ℝ}^{2n}$ with ${ℂ}^n$ and we have asubmanifold 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 foreach$p\in N$ 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 propertyfor 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 disjointunion of submanifolds of different dimensions. It is not hard to see thatif $N$ 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 $N$ tobe the one determined bythe 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 wesay 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 allsubmanifolds $N\subset M$ such that $p\in N$ where we say $N_1$ isequivalent to $N_2$ if there is some open neighbourhood $U$ of $p$ suchthat $N_1\cap U=N_2\cap U$ 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 hypersurfaceas it is of codimension 1.