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I'm following my first course in geometry and we're using the book "Geometry and Topology" by Miles Reid and Balázs Szendröi. Link

Now, the reason the book is chosen is because my professor believes it to be a great intuition builder. I however, care much more for formality than intuition and the proofs provided in the book are below the quality which is expected of us.

For the course, we'll only be delving in to chapters 1 through 5, but I would like a formal book on the subject. I've been looking at the questions previously asked, but I haven't been able to figure out which books would cover the same material.

Mitchell Faas
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    The most formal you can get is probably Euclid's Elements itself. Here is the link: https://www.amazon.com/Euclids-Elements-Euclid/dp/1888009195 . As far as intuition goes, never sacrifice it for formality. It is in the details (here I mean, naive shuffling of equations) that we sometimes lose sight of the big picture. Your professor has his mind in the right place. Build intuition and then edge out the details. Be patient. – Faraad Armwood May 08 '17 at 19:07
  • I'm doubtful euclids elements includes such things as affine mappings on hyperbolic spaces. When it comes to intuition: I believe there's nothing wrong with a good intuition, but at its core, math lies in definitions. No proof should rely on intuition to get its point across. Sure, it makes coming up with proofs a lot easier, but going for formality gives you the needed abstractions to develop your own intuition, which often gives a much better view of the circumstances. A bit like the copy of a copy ordeal. – Mitchell Faas May 08 '17 at 20:09
  • If you truly want to understand hyperbolic geometry and not just "intuitively" as you assert, you'll need to know a good bit of differential geometry and complex analysis. In reference to other users who may comment, you should edit your title since as it stands, it seems as though you want a reference for pure geometry. I will not argue about mathematical philosophy, but I'll just say, trust in your instructor. My advisor has done the same with me and I'm seeing a much greater progression in my studies. – Faraad Armwood May 08 '17 at 22:30
  • As far as a reference to a book, try Elementary Differential Geometry by Presley and Geometry of Surfaces by Stilwell. The latter has everything you need. – Faraad Armwood May 08 '17 at 22:38
  • I am not convinced Euclid's Elements are "the more formal"; IMHO, their interest is only historical. In case one wants to read them, here is a site where they can be consulted, for example for book III: (http://aleph0.clarku.edu/~djoyce/java/elements/bookIII) – Jean Marie May 09 '17 at 01:05
  • Have a look at (https://math.stackexchange.com/q/107882). – Jean Marie May 09 '17 at 01:08
  • I believe what you're saying is that at times the proofs are incomplete in this book. I haven't read it, but based on a cursory glance at its table of contents, it would seem to be quite wide-ranging. It might be best if you singled out some topics and/or theorems in the book where you feel the presentation is incomplete. Also, what passes for an "introduction" to geometry will depend heavily on the background in algebra, topology and analysis assumed on the part of the reader. A wide-ranging book with high prerequisites in those areas is Geometry by Marcel Berger. A further question is... – user49640 May 09 '17 at 05:18
  • whether you would accept a book that takes linear algebra and coordinates as its basis and builds from there, or a book that works from a set of axioms similar in principle to Euclid's (for Euclidean or non-Euclidean geometry), but with a modern and fully rigorous approach. – user49640 May 09 '17 at 05:20
  • user49640: The course is supposed to be taken before any course in analysis or topology. So linear algebra and basis is fine. But even then I like an axiomatic approach where possible.

    I would extract a proof from the book and add them to the post, but I haven't been able to find a single (non linear algebra) proof that does not use imagery to make its argument.

     – Mitchell Faas May 09 '17 at 08:48
  • Proofs that use intuitive-sounding arguments can be acceptable in certain cases, provided it's immediately apparent how to turn them into rigorous arguments. That depends partly on the experience of the reader, however. – user49640 May 09 '17 at 15:09
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    I can't suggest a book that will cover everything, but rigorous books with a formal style that use linear algebra as their basis include Linear Algebra and Geometry by Dieudonné, Geometry: A Comprehensive Course by Pedoe, and Cours de Géométrie by Tauvel. Rigorous books that start from Euclid-style axioms are Foundations of Geometry by Hilbert (this was the first-ever fully rigorous exposition), Géométrie euclidienne plane by Doneddu, Higher Geometry by Efimov, Lectures on the Foundations of Geometry by Pogorelov. There is also the book Geometry: Euclid and Beyond by... – user49640 May 09 '17 at 15:17
  • Hartshorne. That book basically asks you to read parts of Euclid's Elements in its first couple of chapters, and makes various comments on it. Then the book shows how Hilbert made Euclid's approach rigorous and explores various issues in the foundations of Euclid-style systems. However, the book expects a relatively high level of mathematical maturity on the part of the reader, and therefore the proofs may not seem rigorous or complete to someone who doesn't have enough experience to fill them in. It's much less tedious than reading Hilbert's book, though. – user49640 May 09 '17 at 15:22
  • You might also be interested in Coxeter's Non-Euclidean Geometry. – user49640 May 09 '17 at 15:50

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