Differential forms on manifolds

Differential forms on manifolds

We have been thinking of differential forms on \mathbb{R}^n. Hubbard defined them as follows:

Definition. A k-form on \mathbb{R}^n is a function \phi that takes k vectors in \mathbb{R}^n and returns a number, such that \phi(v_1, \ldots, v_k) is multilinear and antisymmetric.

I want to extend this notion to work on manifolds.

Let’s use less patronising language then, and call a k-covector of a vector space V, a k-linear map f:V \times \cdots \times V \rightarrow \mathbb{R} satisfying

f(v_{\sigma(1)}, \ldots, v_{\sigma(k)}) = sgn(\sigma)f(v_1, \ldots, v_k)

for \sigma \in S_k the symmetric group. (Make sure you know what k-linear means)

Exercise.

  1. Check that the space of k-covectors of V, denoted A_k(V), is a vector space.
  2. If x_1, \ldots, x_n is a basis for V, make sure you know why

\{\alpha_{i_1} \wedge \cdots \wedge \alpha_{i_k} , 1\leq i_1 < \cdots < i_k \leq n\}

is a basis for A_k(V), where \alpha_i is the 1-covector that sends x_i to 1, and x_j to 0 for all j \neq i. Remember, the wedge symbol \wedge, is just formal notation to remind you that the covector is antisymmetric.

Observe that setting V = \mathbb{R}^n in the definition of a k-covector returns precisely the definition Hubbard gave of a k-form on \mathbb{R}^n. The idea now is to extend this to general manifolds which we do by considering V = T_pM the tanget space.

Definition. A k-form on a manifold M is a function \omega that assigns to each point p \in M a k-covector \omega_p in A_k(T_pM)

So at each point on our manifold, we have a map that takes vectors in the tangent space T_pM to \mathbb{R}. This is what I mean in class when I talk about sections of the cotangent bundle.

Remark. In mathematics we have the notion of a dual. Given a vector space V with basis e_1, \ldots, e_n, we have its dual vector space, denoted, V^\ast with basis \alpha_1, \ldots, \alpha_n where, as above, \alpha_i(e_j) = \delta_{ij} (the Kronecker delta function). It is common to use the prefix “co” to talk about the dual concept, for instance, v \in V is a vector, \phi \in V^\ast is a covector. In the same way we have the tangent space T_pM and the cotangent space, denoted T^\ast_pM. To that end, let’s bring our notation above in line with the prevailing convention and denote

\displaystyle A_k(T_pM) = \bigwedge^k(T_p^\ast M)

(Exercise. What should the relationship be with (\bigwedge^k(T_pM))^\ast?)

Example.  Okay – well \mathbb{R}^n is a manifold (someone even claimed it was their favourite manifold!), so we better make sure the two notions we have agree. Luckily the situation here is rather simple as the tangent space at a point in \mathbb{R}^n is isomorphic to \mathbb{R}^n, so there is nothing to do. We can do better and give a description of k-forms on \mathbb{R}^n that is more in line with general manifolds. At each point p \in \mathbb{R}^n  we have a standard basis for the tangent space T_p\mathbb{R}^n,

\displaystyle\left.\frac{\partial}{\partial x_1}\right|_p, \ldots, \left.\frac{\partial}{\partial x_n}\right|_p

Let dx_1|_p, \ldots, dx_n|_p be the dual basis, that is, dx_i|_p(\frac{\partial}{\partial x_1}|_p) = \delta_{ij}. Then we have a basis for the vector space A_k(T_p\mathbb{R}^n) = \bigwedge^k (T^\ast_p \mathbb{R}^n)

\{dx_{i_1}|_p \wedge \cdots \wedge dx_{i_k}|_p : 1 \leq i_1 < \cdots < i_k \leq n \}

And so a k-form \omega on \mathbb{R}^n, is a function \omega: \mathbb{R}^n \rightarrow \bigwedge^k(T^\ast \mathbb{R}^n) sending p \mapsto \omega_p of the form,

\omega_p=\sum a_{i_1\cdots i_k}(p) dx_{i_1}|_p\wedge\cdots\wedge dx_{i_k}|_p, where a_{i_1\cdots i_k} is a function of \mathbb{R}^n

So for example, \omega = (\sin x_1\cdot \cos x_3) ~ dx_1 \wedge dx_2 is a 2-form on \mathbb{R}^3, which takes p = (p_1, p_2, p_3) to \omega_p = (\sin p_1 \cdot\cos p_3) ~ dx_1|_p \wedge dx_2|_p

The situation on a general manifold M is not much harder, since we have a system of local coordinates for M. The standard way to do this is to consider a system of charts on your manifold, giving you your local coordinates, but I’ll save that for another post as this one is already getting a bit long. The upshot is that for a manifold we can define a k-form \omega as a function taking points p \in M to k-covectors \omega_p \in \bigwedge^k(T^\ast_pM).

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