0708-1300/Class notes for Tuesday, November 6
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Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
We will now shift our attention to the theory of integration on smooth manifolds. The first thing that we need to construct is a means of measuring volumes on manifolds. To accomplish this goal, we begin by imagining that we want to measure the volume of the "infinitiesimal" parallelepiped [1] defined a set of vectors [math]\displaystyle{ X_1 , \ldots ,X_k \in T_pM\! }[/math] by feeding these vectors into some function [math]\displaystyle{ \omega : (T_pM)^k \to \mathbb{R}\! }[/math]. We would like [math]\displaystyle{ \omega\! }[/math] to satisfy a few properties:
- [math]\displaystyle{ \omega\! }[/math] should be linear in each argument: for example, if we double the length of one of the sides, the volume should double.
- If two of the vectors fed to [math]\displaystyle{ \omega\! }[/math] are parallel, the volume assigned by [math]\displaystyle{ \omega\! }[/math] should be zero because the parallelepiped collapses to something with lower dimenion in this case.
Inspired by these requirements, we make the following definition:
Definition
Let [math]\displaystyle{ V\! }[/math] be a real vector space, let [math]\displaystyle{ p \in \mathbb{N}\! }[/math] and let [math]\displaystyle{ L(V^p; \mathbb{R}) }[/math] denote the collection maps from [math]\displaystyle{ V^p\! }[/math] to [math]\displaystyle{ \mathbb{R} }[/math] that are linear in each argument separately. We set
[math]\displaystyle{ A^p(V) = \left\{ \omega \in L(V^p; \mathbb{R}) : \omega(\ldots,v,\ldots,v,\ldots) = 0\ \forall v \in V \right\} }[/math]
Proposition
Suppose that [math]\displaystyle{ \omega \in A^p(V)\! }[/math] and [math]\displaystyle{ v_1,\ldots,v_p \in V\! }[/math]. The following statements hold:
- [math]\displaystyle{ A^p(V)\! }[/math] has a natural vector space structure
- [math]\displaystyle{ A^0(V)\! }[/math] is [math]\displaystyle{ \mathbb{R} }[/math]
- [math]\displaystyle{ A^1(V) = V^*\! }[/math] is the dual space of [math]\displaystyle{ V\! }[/math]
- [math]\displaystyle{ \omega(v_1,\ldots,v_j,\ldots,v_k,\ldots,v_p) = - \omega(v_1,\ldots,v_k,\ldots,v_j,\ldots,v_p)\! }[/math] for every [math]\displaystyle{ j\lt k \in \{1,\ldots,p\}\! }[/math]
- If [math]\displaystyle{ \sigma \in S_p\! }[/math] is a permutation, then [math]\displaystyle{ \omega(v_{\sigma(1)},\ldots,v_{\sigma(p)}) = (-1)^\sigma \omega(v_1,\ldots,v_p) }[/math]
Proof
The first statement is easy to show and is left as an exercise. The second statement is more of a convenient definition. Note that [math]\displaystyle{ A^0(V)\! }[/math] consists of all maps that take no vectors and return a real number since the other properties are vacuous when the domain is empty. We can thus interpret an element in this space simply as a real number. The third statement is clear as the defintions of [math]\displaystyle{ A^1(V)\! }[/math] and [math]\displaystyle{ V^*\! }[/math] coincide.
As for the fourth, note that [math]\displaystyle{ 0 = \omega(v_1, \ldots, v_j + v_k, \ldots, v_j+v_k, \ldots, v_p) }[/math] so that using linearity we obtain
[math]\displaystyle{ 0= \omega(v_1, \ldots, v_j , \ldots, v_j, \ldots, v_p) +\omega(v_1, \ldots, v_k , \ldots, v_j, \ldots, v_p) + \omega(v_1, \ldots, v_j , \ldots, v_k, \ldots, v_p) +\omega(v_1, \ldots, v_k , \ldots, v_k, \ldots, v_p) }[/math]
and hence [math]\displaystyle{ \omega(v_1, \ldots, v_k , \ldots, v_j, \ldots, v_p) + \omega(v_1, \ldots, v_j , \ldots, v_k, \ldots, v_p) = 0 }[/math].
The fifth statement then follows from repeated application of the fourth.
Remarks
Our computation in the previous proof shows that we could equally well have defined [math]\displaystyle{ A^p(V)\! }[/math] to consist of all those multilinear maps from [math]\displaystyle{ V^k\! }[/math] to [math]\displaystyle{ \mathbb{R}\! }[/math] that change sign when two arguments are interchanged.
One of the nicest things about these spaces is that we can define a sort of multiplication of elements of [math]\displaystyle{ A^p(V)\! }[/math] with [math]\displaystyle{ A^q(V)\! }[/math]. This multiplication is called the wedge product and is defined as follows.
Definition
For each [math]\displaystyle{ p,q \in \mathbb{N}\! }[/math] the wedge product is the map [math]\displaystyle{ \wedge : A^p(V) \times A^q(V) \to A^{p+q}(V), (\omega,\lambda) \mapsto \omega \wedge \lambda }[/math] defined by
[math]\displaystyle{ (\omega \wedge \lambda) (v_1,\ldots,v_{p+q}) = \sum_{\sigma \in S_{p+q}} \frac{(-1)^\sigma}{p!q!} \omega(v_{\sigma(1)},\ldots,v_{\sigma(p)})\lambda(v_{\sigma(p+1)},\ldots,v_{\sigma(p+q)}) }[/math]
for every [math]\displaystyle{ v_1 ,\ldots,v_{p+q} \in V }[/math].
