# 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 ${\rho}_{k}:U\to \mathbb{R}$ where the differentials of ${\rho}_{k}$ are linearly independent^{},
such that

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

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 ${\u2102}^{n}$ and we have a submanifold there it is called a real submanifold in ${\u2102}^{n}$. ${\rho}_{k}$ 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 $\phi :U\to {\mathbb{R}}^{m}$ 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 ${\mathbb{R}}^{n}$. This is in fact a hypersurface as it is of codimension 1.

## References

- 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
- 2 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.

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 |