**Week 6**

1. The Cotangent bundle of an *m*-manifold *M*.

A. Definition *via* transition functions. Consider *M* together with
a representative atlas {*h*_*a*: *U*_*a* -->
R^*n*} of coordinate charts. We have
seen that setting *g*_*ab*(*x*) = (D*x*^*a*_*i*
/D*x*^*b*_*j*)(*x*) defines a Gl(*m*,R)-valued
cocycle, and therefore an* *m-dimensional vector bundle which we have
identified with the tangent bundle.

(Remember that *x*^*a*_*i*
= *x*_*i* o *h*_*a* =
the *i*-th component of the *h*_*a* coordinate,
and that D*f*/D*x*^*b*_*j*(*x*) =
D/D*x*_*j*
(*f* o *h*_*b*^(-1))(*h*_*b*(*x*)).)

Defining *g**_*ab* = *g*_*ba*^T (transpose)
gives a new cocycle, as can easily be checked. The corresponding
vector bundle is the cotangent bundle.

B. Definition *via* covectors. A *covector* at a point
*x* in *M* is a linear map *omega* : T*M*_*x* --> R.
The set of such linear
maps is the vector space T**M*_*x*, the co-tangent
space at *x*. For
example, if *f* is a smooth function defined near *x*,
then *f* defines
the covector d*f*_*x*(*v*) = *v*(*f*).
In particular, if *x* is in the
coordinate chart *h*_*a*: *U*_*a* --> R^*m*,
the *m* coordinate functions
*x*^*a*_*i* = *x*_*i* o *h*_*a*
define covectors d*x*^*a*_1,..., d*x*^*a*_*m*.
These covectors
form a basis for the cotangent space: if
*omega*(D/D*x*^*a*_*i*)
= *A*^*i*,
then *omega* =
sum_*i* *A*^*i* d*x*^*a*_*i*.
This allows the identification
of T**U*_*a* with *U*_*a* x R^*m*.
In particular d*f* = sum_*i* D*f*/D*x*^*a*_*i*
d*x*^*a*_*i*.
Applying this last formula to the coordinates *x*^*b*_*1*
,..., *x*^*b*_*m* of
another chart *h*_*b*: *U*_*b* --> R^*m*
defined near *x* gives
d*x*^*b*_*j* =
sum_*i* D*x*^*b*_*j*/D*x*^*a*_*i*
d*x*^*a*_*i*.
So if *omega* = sum_*j* *B*^*j*
d*x*^*b*_*j*
then *omega* = sum_*j* *B*^*j* sum_*i*
D*x*^*b*_*j*/D*x*^*a*_*i*
d*x*^*a*_*i*;
comparing coefficients of d*x*^*a*_*i*
yields *A*^*i* = sum_*j* *B*^*j*
D*x*^*b*_*j*/D*x*^*a*_*i*,
so the transition function relating the local trivializations
over *U*_*a* and *U*_*b* is
*g**_*ab* = *g*_*ba*^T.

2. Vectorfields and 1-forms

*Definition* If *p*: *E* --> *B* is a (vector) bundle, a
*smooth section* in *E* is a smooth map *s*: *B*
--> *E*
such that *p* o *s* = id_*B*. A section ``picks out a point in
each fiber'' in a smooth way.

*Definition* A smooth section in T*M* --> *M* is a
smooth *vectorfield*; a smooth section in T**M* --> *M*
is a smooth covectorfield, or more usually a *smooth*
1-*form*.

The bundles T*M* and T**M* are
topologically equivalent (see homework)
but not in any natural way. There is an important law of
composition between smooth vectorfields, called the *Lie*
*bracket*, defined by [*X*,*Y*]*f* =
*X*(*Y**f*)-*Y*(*X**f*). Class exercise:
show that [*X*,*Y*] is indeed a smooth vectorfield if *X*
and *Y* are.
But this law is non-associative, and does not permit useful
computations with larger sets of vectorfields. We will now
study an operation on forms; it is not a law of composition
between covectorfields because the ``product'' of two 1-forms is
a new kind of object called a 2-form.

First some multilinear algebra. Given a real vector space *V*, a
*linear* *p*-*form* on
*V* is a multilinear, alternating map
*K*: *V* x ... x *V* --> R (*p* factors of *V*).
Special cases: a linear
1-form is a linear map *V* --> *R*; if *V* has dimension
*n*, then
*X*_1 ,..., *X*_*n* --> det[*X*_1 ... *X*_*n*]
(the determinant of the
matrix whose *i*th row is *X*_*i*) is well known to be
*n*-linear
and alternating; a standard fact from linear algebra says
that *any* *n*-linear alternating function
*V* x ... x *V* --> R (*n* factors)
is some constant multiple of the determinant; so the vector
space of linear *n*-forms on *V* is 1-dimensional.
Clearly the vector space
of linear 1-forms is *n*-dimensional.

{Terminological note: I am using ``linear *p*-form'' here to distinguish
this object from a smooth *p*-form on a manifold *M*,
which will be a section in the bundle T*^*p**M* --> *M*
with fibre over *x* in *M* the vector space of
linear *p*-forms on T*M*_*x*. Usually both are called
simply ``*p*-form.''}

Proposition. If *V* has dimension *n*, the dimension of the vector
space of linear *p*-forms on *V* is the binomial coefficient
C(*n*,*p*).

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