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I'm looking for a recommendation for a book that has a down-to-earth yet informative introduction to tensors, manifolds etc. for general relativity. Some context:

I'm studying GR this term. Outside of class, I intend to finish studying the theory of multivariable real-valued functions so I can start off with manifold theory at the start of the summer. So for the time being, I am looking for a textbook that I can use to supplement my GR class and introduce myself to manifolds, tensors etc. before I start covering these topics rigorously hopefully in a few months' time.

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    *Tensor Geometry: The Geometric Viewpoint and its Uses* by C. T. J. Dodson and T. Poston (GTM series) is quite readable – 2017-01-27

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I would recommend the three books by John M. Lee which are very well written with alot of exercises helping you understanding the material better. I read a lot of books and rarely you get the feeling Hmm...this definition or theorem is very abstract. For what can I need it? How does it look in a more concrete case like metric spaces? And then just reading the last line of the paragraph there is an embedded exercise which just answers your question. Perfect! So my recommendation list is the following: - Introduction to topological manifolds. Contains basic properties and usefull theorems on topological spaces (maybe as refresher) and then goes straight into (topological) manifold theory. You may skip a larger portion of the book since it treats surfaces extensively and very well. - Introduction to smooth manifolds. This book contains basic analysis on differentiable manifolds, introducing vector fields, tensors, Lie groups and so on. So a basic yet extensiv introduction to differential geometry. - Riemannian manifolds. This just might be very supplementary. The first section of the book does study again tensors, it has a few more exercises to do.

So my summary:

I would start with introduction to topological manifolds to grasp on the ideas of topology and seeing the definition of a manifold. Then it is usefull to know that manifolds can be equipped with different structures, in your case smooth or differentiable manifolds. So you can read some pages in introduction to smooth manifolds. If you wish rigorous geometry, Riemannian manifolds serves its purpose.

Also it was usefull for me to have this books during my studies of topology (mainly the one for topological manifolds) and I will surely use them later for references.

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    I was planning to take this route AFTER I have studied the theory of multivariable functions. Don't you think it'd be premature for me to start studying these books? – 2017-01-27
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    @JunaidAftab The thing is what I have learned in mathematics is that sometimes you have get aware of the fact that there are really difficult topics and its always getting more difficult but with time you start getting used to it. So especially with things you requested it is important to learn the basic definitions early, so you get used to it. Especially GR is not down to earth since it is based on advanced concepts. Hence a down to earth treatment of this would just be a complification of it (e.g. see the book by Bloch). Then you land in intermediate level which does not help you at all. – 2017-01-27
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    Perhaps I should restate my motivation for asking the question: I'm very much interested in going through these books, but I suppose I will be rushing through them at these points. At this point, I am looking for textbooks that'll introduce me to these concepts at the intermediate level, so I can see the material being presented in GR at a more firm grounding. My motivation is that eventually in a few months once I have finished the course on GR, I will be able to tackled advanced book and further broaden my understanding. Ergo, I was hoping to look for books that'll make me make this... – 2017-01-27
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    transition, while in the mean time I cover my basics: theory of multivariable functions, doing a bit of algebra which I haven't done and revising on my point set topology. – 2017-01-27
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    @JunaidAftab As I said it is very difficult to introduce such advanced topics at intermediate level. It brings nothing. But ok I see. If you wish to revise point set topology and also learning something about manifolds I would recommend to go through chapter $2$, $3$ and $4$ of introduction to pological manifolds (around $100$ pages, which is doable). I hope this more concrete suggestion helps you. See my answer for later. – 2017-01-27
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    I just skimmed through the relevant sections of the boon introduction to topological manifolds. My concerns are that on a first look, I get the impression that the few first chapters revise basic properties in point-set topology and introduce manifolds. Again by your reasoning, wouldn't this be insufficient as well: I'm not learning anything new or substantial if I merely restrict myself to these units. Hence I asked the question: prior to working these books in detail, it would be awesome if I could get a book that'll a) introduce me to such ideas which I can supplement in my GR class and ii) – 2017-01-27
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    prepare me to tackle such books as well. – 2017-01-27