I am reading an appendix on jet bundles, and I am confused on the following question. The note I am reading (Singularity of Mappings by Mond and Nuno-Ballesteros, Appendix A) says the following:
There is also a jet bundle $J^k(X, Y)$ over any pair of (complex or real) manifolds $X$ and $Y$ , whose fibre over $(x,y)\in X\times Y$, which we denote by $J^k(X, Y )_{(x,y)}$ , is the set of $k$-jets of germs of maps $(X,x)\to (Y,y)$.
My question is, how is this jet bundle $J^k(X, Y)$ defined ? And why this is a locally trivial fiber bundle? I think this is a principle $G$ bundle, where $G$ denotes the group of coordinate changing of $\mathbb{C}^n\times \mathbb{C}^p$. How should I understand this?