Next: Integration again
Up: Differential forms.
Previous: Integration of differential forms
  Contents
Let
be a real vector space of
dimension
. Then define
.
This is a real, one dimensional vector space.
So the set
is disconnected.
An orientation of the vector space
is a choice
of one of these connected components.
If
is an invertible linear map from
to
then it induces a linear map from
which is therefore multiplication by a complex number.
This number is just
the determinant of
If
is a manifold of dimension
then the same
applies to
;
is a disconnected set.
We define
Definition 5.2
A manifold is orientable if there is a non-vanishing
![$ n$](img164.gif)
-form on
![$ M$](img54.gif)
. Otherwise it is called non-orientable.
If
and
are two non-vanishing
forms then
for some
function
which is either strictly negative
or strictly positive. Hence the set of
non-vanishing
forms divides into two sets. We
have
Definition 5.3 (Orientation)
An orientation on
![$ M$](img54.gif)
is a choice of one of these two sets.
An orientation defines an orientation on each
tangent space
. We call an
form positive
if it coincides with the chosen orientation negative
otherwise. We say a chart
is
positive or oriented if
is positive. Note that if a chart is not positive we
can make it so by changing the sign of one co-ordinate function
so oriented charts exits. If we chose two oriented charts
then we have that
is an oriented diffeomorphism. The converse is also
true.
Proposition 5.6
Assume we have a covering of
![$ M$](img54.gif)
by co-ordinate
charts
![$ \{(U_\alpha , \psi_\alpha )\}_{\alpha \in I}$](img634.gif)
such that for any
two
![$ (U_\alpha, \psi_\alpha)$](img148.gif)
and
![$ (U_\b , \psi_\b )$](img668.gif)
the diffeomorphism
is orientation preserving. Then there is an orientation
of
![$ M$](img54.gif)
which makes each all these charts
oriented.
Proof.
Notice that the fact that
is an oriented diffeomorphism means that if
![$ x\in U_\alpha \cap U_\b $](img671.gif)
then
is a positive multiple of
Hence if
![$ \phi_\alpha $](img635.gif)
is a partition of unity then
is a non-vanishing
![$ n$](img164.gif)
form. Clearly this defines the
required orientation.
Next: Integration again
Up: Differential forms.
Previous: Integration of differential forms
  Contents
Michael Murray
1998-09-16