Exponential of a 2-form

Question

What does $e^{\omega}$ means when $\omega$ is a $2$-form? Is it a $2$-form again? If it is a $2$-form, is its definition $\displaystyle e^{\omega}(u,v)=\sum_{n=0}^{\infty} \frac{\omega(u,v)^n}{n!}$?

Answer 1

This isn't antisymmetric in $u,v$, the problem comes from the even terms. So this doesn't define a $2$-form. You could however define the exponential of $\omega$ using the standard formula (it's a finite sum in any case) $$\sum_{i=0}^\infty \frac{\omega^n}{n!}=1+\omega+\frac{\omega^2}{2}+\cdots+\frac{\omega^p}{p!}$$ where $p$ is defined so that $2p$ is the largest integer $\leq\dim(M)$. This defines a differential form distributed in even degrees.