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 as that’s the most useful case.
If are in fact smooth then is a smooth submanifold and similarly if is real analytic then is a real analytic submanifold. If we identify with and we have a submanifold there it is called a real submanifold in . are usually called the local defining functions.
Let’s now look at a more general definition. Let be a manifold of dimension . A subset is said to have the submanifold property if there exists an integer , such that for each there is a coordinate neighbourhood and a coordinate function of such that , if or if .
Let be a manifold of dimension . A subset with the submanifold property for some is called a submanifold of of dimension and of codimension .
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 is connected this is not an issue (whatever the topology on is).
In case of differentiable manifolds, if we take to be a subspace of (the topology on is the relative topology inherited from ) and the differentiable structure of to be the one determined by the coordinate neighbourhoods above then we call a regular submanifold.
If is an open subset of , then is called the open submanifold of . This is the easiest class of examples of submanifolds.
Example of a submanifold (a in fact) is the unit sphere in . This is in fact a hypersurface as it is of codimension 1.
- 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.
|Date of creation||2013-03-22 14:47:20|
|Last modified on||2013-03-22 14:47:20|
|Last modified by||jirka (4157)|
|Defines||codimension of a manifold|
|Defines||local defining functions|
|Defines||real analytic submanifold|
|Defines||germ of a submanifold|