I'm studying Classical Mechanics, and in some cases the particles in the study are constrained to move in a certain manifold, which changes with time.
I've looked for bibliography about time-depending manifolds, or something similar I could apply to this study, but I haven't found anything yet, so I'd greatly appreciate any bibliographical suggestion.
Any book dealing with non-autonomous differential equations (so pretty much any book) should fit the bill. If you want better understanding of just manifolds Tu's Introduction to Manifolds is a reasonable choice. For ODEs I would recommend Ordinary Differential Equations with Applications by Chicone (very theoretically rigorous and does most of its work in Banach Spaces). For PDE, one text to consider is Taylor's Partial Differential Equations: basic theory (Springer). If you want a treatment specifically in the language of manifolds, and taking a more differential geometry approach Pressley's Elementary Differential Geometry (Springer) is a pretty solid (and reasonably priced) introduction.