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I am a master's student with interests in algebraic geometry and number theory. And I have a good collection of textbooks on various topics in these two fields. Also, as part of my undergraduate curriculum, I learnt abstract algebra from the books by Dummit-Foote, Hoffman-Kunze, Atiyah-MacDonald and James-Liebeck; analysis from the books by Bartle-Sherbert, Simmons, Conway, Bollobás and Stein-Shakarchi; topology from the books by Munkres and Hatcher; and discrete mathematics from the books by Brualdi and Clark-Holton. I also had basic courses in differential geometry and multivariable calculus but no particular textbook was followed. (Please note that none of the above-mentioned textbooks was read from cover to cover).

As you can see, I didn't learn much geometry during my past 4 years of undergraduate mathematics. In high school, I learnt a good amount of Euclidean geometry but after coming to university geometry appears very mystical to me. I keep hearing terms like hyperbolic/spherical geometry, projective geometry, differential geometry, Riemannian manifold etc. and have read general maths books on them, like the books by Hartshorne, Ueno-Shiga-Morita-Sunada and Thorpe.

I will be grateful if you could suggest a series of books on geometry (like Stein-Shakarchi's Princeton Lectures in Analysis) or a book discussing various flavours of geometry (like Dummit-Foote). I am aware that Coxeter has written a series of textbooks in geometry, and I have read Geometry Revisited in high school (which I enjoyed). If these are the ideal textbooks, then where to start? Also, what about the geometry books by Hilbert?

Thank you for reading.

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    $\begingroup$ Look up John Lee's texts: Introduction to Topological Manifolds and Introduction to Smooth Manifolds. These might be of interest to you. $\endgroup$ – Clayton Mar 24 '18 at 6:02
  • $\begingroup$ @Clayton I looked at Lee's Introduction to Topological Manifolds at the library. I jumped to the chapter on CW complexes and I liked it (it's much better than Hatcher). Thanks. $\endgroup$ – rationalbeing Mar 24 '18 at 10:26
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    $\begingroup$ When I think “reference request related to algebraic geometry” I think “Javier Álvarez”. See this answer for instance (and many of his other reference-request answers contain good things). I don’t think it’s mentioned in that answer, but Shaferavich has a book or two on geometry that may be worth looking at. $\endgroup$ – pjs36 Mar 24 '18 at 14:39
  • $\begingroup$ For differential geometry: Noel Hicks' "Notes on Differential Geometry" and Milnor's "Morse Theory". Also, books on general relativity can be good ways to learn some geometry (e.g. Carroll's "Spacetime and Geometry"). $\endgroup$ – Yly Mar 24 '18 at 17:49
  • $\begingroup$ @pjs36 Thanks for sharing Javier Álvarez's nice answer. $\endgroup$ – rationalbeing Mar 24 '18 at 17:50
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Geometry is really a very broad term and it encompases many different realms of mathematics.

Usually, the main dichotomy is between algebraic geometry and differential geometry.

For algebraic geometry, 2 series come to mind:

1.Kenji Ueno, Algebraic Geometry (these are 3 monographs by AMS) 2.Cox, O'Shea , Little: 1)Ideals, Varieties, and Algorithms/2) Using Algebraic Geometry

Also there are the vast notes by Ravi Vakil on his website, which I highly recommend (I think that studying Ueno and then Vakil will give you a very good foundation in algebraic geometry).

Now for Differential Geometry again 2 series come to mind:

  1. John Lee: Topological Manifolds/Smooth Manifolds/Curvature
  2. Tu: An Introduction to Manifolds/ Differential Forms in Algebraic Topology (allong with Bott)/Connections, Curvature and Characteristic Classes.

Also, one could start reading Guillemin & Pollack: Differential Topology and then the classical Milnor: Topology from the differential viewpoint.

Finally, you could also take a look in Berger: A panoramic view of Riemannian Geometry where the author aims tou give a quick but to the point description of all the areas of that vast subject.

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    $\begingroup$ I will look at Ueno's books. Also, do you mean Guillemin and Pollack: Differential Topology? I couldn't find any Differential geometry book by them. $\endgroup$ – rationalbeing Mar 24 '18 at 10:30
  • $\begingroup$ @rationalbeing Yes, you are right, I will edit my answer. $\endgroup$ – Nick A. Mar 24 '18 at 10:58
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    $\begingroup$ Nick A., are you sure there's a book by Tu called 'introduction to smooth manifolds' ? all i found was 'introduction to manifolds'. however, there's a book by john lee called 'introduction to smooth manifolds' $\endgroup$ – BCLC Aug 28 '18 at 3:06
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    $\begingroup$ @BCLC ooops you are right I will edit that . Thanks ! $\endgroup$ – Nick A. Aug 31 '18 at 19:19
  • $\begingroup$ @NickA. Thanks too! $\endgroup$ – BCLC Aug 31 '18 at 19:21

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