Notation on differential forms

Question

This is a homework problem. Could anyone tell me the meaning of the symbols $i_{X},i_{Y}$ here? Thanks a lot.

Also, does $\Omega^{*}(M)$ here mean the exterior algebra bundle on the manifold $M$?

Answer 1

The notation $i_X$ here is the interior product, which is just contraction with the first slot: for a $k$-form $\omega$, $i_X \omega$ is the $k-1$-form defined by $i_X \omega(Y_2,\ldots,Y_k) = \omega(X,Y_2,\ldots,Y_k).$

On the second point, you're basically right: $\Omega^*(M)$ usually refers to the space of sections of the exterior algebra of the cotangent bundle, i.e. $\Omega^*(M) = \Gamma(M, \Lambda T^*M)$ where $\Lambda T^*M = \oplus_{k=0}^m \Lambda^k (T^*M).$ That is, $\Omega^*(M)$ is just the space of all (smooth global) differential forms (possibly of mixed order) on $M$.