Here is a list of courses/books one can use. This list is presented in no particular order of reading, but in a order I feel easier to understand.
To begin with
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The most important book is Differential Geometry of Curves and Surfaces by Manfredo doCarmo.
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Associated lecture by Claudio Arezzo at ICTP. The lectures go through the chapters of the book and make the reading so much easier.
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Gravity and Light: A differential geometery course with a taste of physics, spacetime and gravitation. video lectures tutorials
Not to scare anyone, but this is the first lecture's first minute. It is so amazing.
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Differential Geometry for Computer Science by Justin Solomon
- Shapes and Diffeomorphisms by Laurent Younes. link
- Differential Geometry: Curves, Surfaces and Manifold by Wolfgang Kuhnel.
- Introduction to Smooth Manifolds by John M Lee.
- Numerical Geometry of Rigid Shapes by Alex Bronstein, Michael Bronstein and Ron Kimmel.
- Gravitation by Charles W. Misner, John Archibald Wheeler, and Kip Thorne
- A panaromic view of Riemannian Geometry by Marcel Berger. link
- Calculus on Manifolds by Michael Spivak.
- Spacetime and Geometry: An Introduction to General Relativity by Sean Carroll.
- Statistical Shape and deformations by Guoyan Zheng, Shuo Li, Gabor Szekely link A statistical shape analysis book with more focus on medical imaging.
At times, I get lost with certain definitions which has roots in Group Theory. One might encounter Lie Groups and notions of Symmetry sooner or later while going through these material. Instead of going diiging into Abstract Algebra books. Here is a good quick reference material explained on this page Also you can refer to this short definitions
In my opinion, someone interested in geometry should also have a keen interest in physics with emphasis on classical mechanics, Einstein's relativity. It makes the reading much easier as it helps one to visualize the concepts. In that spirit some of the popular science books in no particular order are:
- The Road to Reality by Roger Penrose: link This is an amazing book if you have the time to go through it. It is huge over a 1000 pages. The reason it is such an important book is because it takes you on a chronological journey of mathematics. Understanding the limitations of available mathematical tools will help one to appreciate the efforts made in discovering new tools.
- Special Relativity and Classical Field Theory: The Theoretical Minimum: link This is a book which I still have to read but it follows the lectures of Prof. Leonard Susskind and the amazing website The Theoretical Minimum
- Topology and Geometry: An amazing lecture series by Prof. Tadashi Tokieda on YouTube
- 3Blue1Brown: An amazing YouTube channel discussing Topology and visualizations. Here is the link
- Mathematical Methods and its Applications: If you get stuck on some small mathematical notion which we always do at times. Here is a lecture series which can help link
- Fermat's Last Theorem - The Theorem and Its Proof: An Exploration of Issues and Ideas: An amazin 1993 documentry about Andrew Wiles and his struggles with Fermat's Last Theorem. Here is the movie
- Logicomix: An epic search for truth by Christos Papadimitriou: An amazing comic-like book about the historical context and discussions between Cantor, Russel and Wittgenstein.
- A Mathematician's Apology by GH Hardy: A letter written by Hardy towards the end of his career and his reflections on his mathematical career. Here is the letter.
I am working on it and will be updated soon with key research articles focussed on medical imaging.
- Topology TooKit by Julien Tierny and the associated book Topological Data analysis for scientific viualization