Editing Submanifold (section)

==Properties==

Given any immersed submanifold <math>S</math> of <math>M</math>, the [[tangent space]] to a point <math>p</math> in <math>S</math> can naturally be thought of as a [[linear subspace]] of the tangent space to <math>p</math> in <math>M</math>. This follows from the fact that the inclusion map is an immersion and provides an injection
: <math>i_{\ast}: T_p S \to T_p M.</math>

Suppose ''S'' is an immersed submanifold of <math>M</math>. If the inclusion map <math>i: S\to M</math> is [[closed map|closed]] then <math>S</math> is actually an embedded submanifold of <math>M</math>. Conversely, if <math>S</math> is an embedded submanifold which is also a [[closed subset]] then the inclusion map is closed. The inclusion map <math>i: S\to M</math> is closed if and only if it is a [[proper map]] (i.e. inverse images of [[compact set]]s are compact). If <math>i</math> is closed then <math>S</math> is called a '''closed embedded submanifold''' of <math>M</math>. Closed embedded submanifolds form the nicest class of submanifolds.