Next: Vector fields.
Up: The tangent space.
Previous: The tangent to a
  Contents
If
is a manifold then we can define a
submanifold of
by using the
principal property of submanifolds in
.
Definition 4.4 (Submanifolds.)
We say that a subset
![$ N \subset M$](img434.gif)
is a submanifold
of dimension
![$ d$](img226.gif)
of a manifold
![$ M$](img54.gif)
of dimension
![$ m$](img435.gif)
if for every
![$ x \in N$](img436.gif)
we can
find a co-ordinate chart
![$ (U, \psi)$](img55.gif)
for
![$ M$](img54.gif)
with
![$ x \in N$](img436.gif)
and such that
Just as before we can define co-ordinates
on
by letting
for each
. Similarly we have
Proposition 4.4
The set consisting of all the charts
![$ (U\cap N, \bar\psi)$](img438.gif)
constructed in this manner is an atlas. Moreover it makes
![$ N$](img410.gif)
a manifold in such a way that the inclusion map
![$ \iota_N \colon N \to M$](img440.gif)
defined by
![$ \iota_N(n) = n$](img441.gif)
is smooth.
Because the condition for being a submanifold is local we can
use the inverse function theorem as in Proposition
4.2 to prove
Proposition 4.5
Let
![$ f \colon M \to N$](img408.gif)
be a smooth map
between manifolds of dimension
![$ m$](img435.gif)
and
![$ n$](img164.gif)
respectively.
Let
![$ n \in N$](img442.gif)
and
![$ Z = f^{-1}(n)$](img443.gif)
. Then if
![$ T_zf$](img444.gif)
is onto for all
![$ z \in M$](img445.gif)
the set
![$ Z$](img337.gif)
is
submanifold of
![$ M$](img54.gif)
. Moreover the image of
![$ \iota_Z$](img446.gif)
in
![$ T_zM$](img447.gif)
is precisely the kernel of
![$ T_zF$](img448.gif)
.
Next: Vector fields.
Up: The tangent space.
Previous: The tangent to a
  Contents
Michael Murray
1998-09-16