# Can you find a 2-form not written as the wedge of two 1-forms?

I was under the impression that all 2-forms are the wedge $(\wedge)$ of two 1-forms. Is it possible to have a 2-form that you can't write as $A\wedge B$ with $A,B$ 1-forms?

• The question in your title is the opposite of the question in the body of your thread. Mariano is responding to the latter. – Ryan Budney Oct 6 '10 at 2:25

## 1 Answer

Yes, it is possible. (And you should find an example yourself: I will not deprive you of the joy of finding it :) )

• Would dA^dB+dC^dD (in R^4) be an example? It's a two form but it can't be written as A^B – JimJones Oct 6 '10 at 2:18
• @JimJones: your example works, but you should not be satisfied until you have a proof of this. Just because you try to write it as the wedge of two one-forms and fail is of course not enough... – Pete L. Clark Oct 6 '10 at 2:44
• @Mariano, awesome answer – BBischof Oct 6 '10 at 2:44
• I still just don't understand – JimJones Oct 6 '10 at 12:51
• @JimJones. I assume that $A,B,C,D$ in your first formula are functions... (Maybe I could add to Mariano's answer: simply look at the definition of 2-forms and remember something about linear independence.) – Agustí Roig Oct 6 '10 at 12:52