# 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 $\omega_p \in \bigwedge^k(T^\ast_pM)$.