In the proof of proposition 7.10 of the book of heat kernels and Dirac operators, there is the following statement:
We define $\theta (\xi) = (X_M, \xi)$ for $\xi \in \Gamma(M, TM), $ then $\theta$ is a one form on M invariant under the action of X such that $d_X \theta = |X|^2 + d\theta $, wich is clearly invertible outside the set $M_0(X)= \lbrace |X|^2 =0 \rbrace.$
What is the definition of an invertible differential form ( or an equivariant differential form) ? And in this example What is the inverse of $d_X \theta $ ?