I am following an introductory course on algebraic topology. Here are two definitions we've made:
- Given a smooth $n$-manifold $M$, a function $f : M \to \mathbb{R}$ is smooth iff, for each smooth chart $(U,\phi)$, the map $f \circ \phi^{-1} : \phi(U) \to \mathbb{R}$ is smooth.
- A differential form on $U \subset \mathbb{R}^n$ is an element of $\mathcal{C}^{\infty}(U) \otimes_{\mathbb{R}} \Omega^*$, where $\Omega^* := \bigwedge \mathbb{R}^n$ is the exterior algebra on $\mathbb{R}^n$; in other words, it smoothly associates an element of the algebra $\Omega^*$ to each point of $U$.
However, we were told that the following definition for a differential form on a smooth manifold $M$ — which is perhaps the straightforward way to synthesize the two above definitions — is wrong:
- A differential form on $M$ is a map $\omega : M \to \Omega^*$ such that in each smooth chart $(U,\phi)$, the map $\omega \circ \phi^{-1} : \phi(U) \to \Omega^*$ is a differential form on $\phi(U)$.
The reason given was that "the exterior derivatives in each smooth chart don't necessarily agree"; that is, it might be the case that for some smooth charts $(U,\phi),(V,\psi)$ and a point $y \in U \cap V$, we might have $(d(w \circ \phi^{-1}))(\phi(y)) \neq (d(w \circ \psi^{-1}))(\psi(y))$. However, I am unable to think of a "differential form" (according to this bad definition) on some manifold for which I can't come up with a "derivative form" which agrees with the actual derivative in each smooth chart. (For instance, it appears to work fine with the manifold being a circle and $\omega$ being the zero-form $\sin \theta$, with two charts giving the angle $\theta$; $d\omega$ would be $\cos \theta \; d\theta$). Does such a counterexample exist? Or is there something else about this "definition" that makes it wrong or unuseful?
(For completeness, the correct definition of a form on $M$ is a collection of forms $\{\omega_{(U,\phi)} \in \Omega^*(\phi(U))\}$ for each chart $(U,\phi)$, with agreement via pullbacks of transition maps.)
Best Answer
It might be more clear here to start with the intrinsic characterization.
We say the exterior algebra of a manifold $\Omega^*M$ is the algebra generated by wedge products of elements of the cotangent bundle $\Omega^*M=\wedge^*T^*M$, or, equivalently, the set of alternating multilinear forms on $TM$. Differential forms are sections of this bundle.
More verbosely, a $k$-form $\omega$ associates each point $p\in M$ with an alternating map $\omega|_p:(T_pM)^k\to\mathbb{R}$ and this map varies smoothly as we vary $p$.
We can, of course, identify a fiber of $\Omega^*M$ with $\wedge^*\mathbb{R}^n$ (by choosing a basis for $T_p^*M$), but there is no canonical way of doing this even locally.
When choosing local coordinates $x^i$ o a neighborhood $U$, the differentials $dx^i$ induce just such a basis for $T^*M$, allowing us to identify $\Omega^*U$ with $\Omega^*\mathbb{R}^n$ locall, i.e. every $k$-form $\omega$ can be written as $$ \omega=\sum_{i_1<\dots<i_k}\omega_{i_1\dots i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k} $$ For $\omega_{i_1\dots i_k}\in C^\infty U$. This frame, however, will depend on the choice of coordinates, subjest to the transformation law $dx^i=\sum_j\frac{\partial x^i}{\partial y^j}dy^j$.
The definition you give is essentially the set of all local coordinate descriptions of the differential form, plus a consistency condition to ensure that they agree.