Can someone recommend me some easy to read books / lecture notes about manifolds? Can someone recommend me some easy to read books or lecture notes (for beginners) about manifolds, local coordinates, differentials etc? Thank you!
 A: I can recommend the following resources as a place to start:


*

*John Lee's Introduction to Smooth Manifolds

*These lecture notes, which concisely cover the basics.

*LW Tu's An Introduction to Manifolds
All of these assume some familiarity with linear algebra, point set topology, differentiation of functions from $\mathbb{R}^n$ to $\mathbb{R}^m$, and the implicit and inverse function theorems.
A: IMHO, the best book for beginners is Milnor's "Topology from a differentiable view point". 
The book is a classic, is short (more or less 50 pages), clear and written by  one of the greatest Matematicians of the last century (he is still alive).
The only problem is that it doesn't address abstract manifolds, for those you will need other books. But if you have to deal only with manifolds embedded in $\mathbb{R}^N$ and you like a geometric approach, Milnor's book is a perfect introduction. 
The book also proves some important topological results as the definition of degree, Hopf theorem and the classification of 1-manifold.
