"[AUG] Phenomenally good intro to Lie groups and Lie algebras via matrix groups. Very concise: about 137 pp ex exercises (no solutions). Prereqs: multivariable calculus & basic analysis; abstract algebra (groups, fields, morphisms, conjugation) and linear algebra. Fast pace, no frills but crystal clear. Must be the best intro by a long shot! Author's AMS webpage has 10th ch on roots and errata."
"[AUG] Extremely concise but well-written. For math majors so be prepared - must be comfortable with abstract algebra among other areas! If you can hack the math, it's a great introduction to math powering the Standard Model of elementary particles!"
"[AUG] Concise, modern introduction geared toward relativity! Having struggled with tensors and been frustrated by the inadequate exposition in relativity books, I was happy to discover this beauty! Check it out via the Search Inside function. Great price too! Cf. Kay's "Schaum's Outline of Tensor Calculus" below."
"[AUG] Solid intro for self-study based on older components approach; covers many topics incl. special relativity. For special and general relativity, learning tensor calculus is a must and this is one of the best - and by far cheapest - book I've found on the subject. My first pick though is Ahsan's "Tensor Analysis with Applications", especially if you're into relativity."
"[G] If you want the real math underlying general relativity, here it is, in a pedagogically brilliant form. Really for mathematicians so be forewarned strong math skills including previous background in differential manifolds required"
"[G] Lucid mathematical development, written in a very readable style. Must know advanced linear algebra (e.g., operators), some abstract algebra e.g. modules), typical classical math for grad physics or math majors. Unique, I think, and really superb."
"[G] Superb intro to HS theory! Clear, careful, concise exposition for those not wanting to fill in important material via exercises, a practice I detest. Prereqs: linear algebra; some analysis, topology, group theory. Nice pace: Hilbert spaces on p. 23; spectrum p.158. Has spectral analysis of unbounded self-adjoint operators. Anyone interested in HS formalism of QM should check it out!"
"[AUG/G] Lucid presentation, concise & self-study friendly. Covers fewer topics than Lee's Intro to Smooth Manifolds and is less detailed. So you won't get "lost in the trees". I used both books to great advantage. Tu is one of my favorites! Make sure you get the new edition (pictured)!"
"[G] Amazing book, best in class for self-study. Anyone wanting to understand the mathematics underlying general relativity would profit from reading this book. It is long but that's because Lee provides a lot of clarifying discussion along with the math. But make sure you have the requisite background first."
"[AUG] Excellent for self-study. Very notation heavy topic so requires perserverance! Many people also recommend Pressley, Elementary Diff. Geo, but overall I preferred this one. Great preparation for more advanced book like Lee, Intro to Smooth Manifolds, or Tu's An Introduction to Manifolds."
"[AUG] Haven't studied this new  contender but from what I've seen via google books or Amazon's search inside function, this looks like a really nice, concise introduction suitable for self-study. Recommend checking it out before deciding on a first book."
"[AUG] Deservedly a classic! Fine for self-study; covers a lot of territory and has answers to odd questions. Recommend first getting some background in topology and metric spaces, not to mention linear algebra and calculus."
"[AUG/G] 3rd edition (1988). A classic and for a good reason! Although it assumes a great deal of mathematical maturity (i.e., ability to deal with a high degree of abstraction), it is very clearly written. I have no idea what more recent abstract algebra books are like but this classic is quite enjoyable to read (well, I haven't quite read it all yet :)."
"[G] This classic used to be way over my head but over the years, I've really come to appreciate it. Amazingly concise and lucid exposition; no fluff, no frills, just math exposition at its very best. For the math mature so read an introductory book first."
"[AUG] Nice first book on real analysis good for self-study. Concise (about 250 pp). Covers set theory basics; real numbers; metric spaces; continuous functions; differentiation; Riemann integration; partial differentiation; multiple integrals. No answers to problems and no Lebesgue integration. Cf. the Amazon "search inside function". A high quality Dover bargain!"
"[G] Exceptionally well-written, concise reference text. Mathematical poetry but definitely def., theorem, proof style. Must be comfortable with both basic linear algebra and abstract algebra (e.g. groups, rings) and that level of abstractness to productively use this book. But if you've paid your math dues, this book is tops. If others confuse me or I don't know something, I look here."
"[AUG] I loved this book: great for making the transition from undergraduate linear algebra to the real stuff. Only drawback is there are no solutions to exercises, which is irritating. For details, please see my Amazon review."
"[AUG] Excellent and thorough introduction, very clear; highly recommended for self-study. Includes solutions to exercises and a handy appendix reviewing necessary mathematical concepts and definitions. For more details, see my Amazon review."
"[AUG] Very clearly written book. Has more than what's needed for a book like Lee, Intro to Smooth Manifolds or other intro differential geometry books. For a refresher, I consult Munkres. But for 1st exposure I recommend Mendelson, which goes from metric spaces to topological spaces. I have the 1st ed (1977)."
"[AUG/G] Very nice topology book for the mathematically more mature at a very nice price. Includes general convergence via theory of nets and theory of filters. Note this book first published in 1970 and so reflects the terminology and style of that time."
"[AUG] Admirably clear, well organized, concise yet has sufficient examples. Covers basics of many key topics but some core topics like separation or metrization theorems aren't in it. Moves from more intuitive metric spaces to abstract topological spaces in a nice way. For 1st exposure I like the organization better than that of Munkres. Reread it for this review and still love it."
"[AUG/G] Tops for self-study but only for the mathematically mature who do not mind an older-style book (and small font). If you're not confident in your math skills, likely not the best choice for 1st exposure. Chock full of fully worked examples. Has short Lessons with good cross-references so they can be read out of sequence. A classic; great buy!"
"[AUG] I found Polking, Boggess & Arnold (1st ed., 2001, ISBN 0-13-598137-9) well-organized, clear and very "well exampled", pace the negative reviews. Not sure why some hate it: my guess - they're simply not ready for "Diffy Q". Requires less math maturity than Tenenbaum. Nice graphics. Fine for self-study. Forget those whimpy whinners and jump on in! 2nd ed might be better but I don't know."
"[UG/AUG] Superb book: clear exposition, beautiful diagrams. This is where I look when I forget something about vector calculus. Note that this is a practical (engineering-oriented) book, for theoretical treatments see a book on analysis such as Rosenlicht or Rudin. I have the 1st ed. (1998)."