# Why are two exterior differential forms on a connected manifold only differed by a function factor?

Suppose $$M$$ is a connected orientable manifold, then if $$\omega,\omega'$$ are two exterior differential-forms giving orientations to $$M$$, then there is a nonvanishing continuous function $$f$$ such that $$\omega'=f\omega$$

Why is this true? I read this statement in the proof of "there exists exactly two orientations on a connected orientable manifold."

Thank you:))

• that's pretty funny. Your title says "deferential" which is a real word. in the question body it is correct "differential" Mar 3, 2022 at 19:19
• @WillJagy Aha, didn't notice that, a copy-paste changes ' i' to ‘e'.
– Eren
Mar 3, 2022 at 19:23
• The best example of auto-correct I've seen is mathoverflow.net/questions/85610/… Mar 3, 2022 at 19:25

The orientation on a manifold is given by a nowhere vanishing top form (form of top dimension).

Now locally a top form looks like $$fdx_{i_{k_{1}}} \wedge ...dx_{{i_{k2}}}$$. Since the orientation form is nowhere vanishing any two such forms can differ by a function that is throughout positive or negative.

This dichotomy implies that there can be only two orientations.

Now try to see that globally also this makes sense if your manifold is connected.

• Thanks, I didn't notice at first that the orientation should be a top form ToT.
– Eren
Mar 3, 2022 at 20:11

A differential form that gives rise to an orientation is of maximal degree and there aren't that many diffenrent differential forms of maximal degree. Consider a single point of the manifold and its tangent space. Let's assume it is of dimension $$3$$. Then there are several different one-forms, say $$dx$$ or $$dy$$. There are also several different two-forms, say $$dx\wedge dy$$ or $$dy\wedge dz$$. But there is only one three-form, namely $$dx\wedge dy \wedge dz$$, up to a real constant.

The same thing happens on the manifold. There is only one differential form of maximal degree at each point, up to a constant. But you can choose a different constant at each point. Hence if you have two different differential forms of maximum degree on a manifold they differ by a function.