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.
Best Answer
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:
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.