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Revision as of 16:00, 6 July 2021
This list of books provides the most direct and rigorous route to understanding differential geometry. Each selection thoroughly addresses its subject matter. There are other books for more specific topics. These are the core. See the image on the right for a visual treatment of its dependencies.
The list does not need to be read linearly or only one book at a time. It is encouraged to go between books and read several at a time to acquire the necessary language and understand the motivations.
The Royal Road to Differential Geometry and Physics is the focus of the list. While on that track, you should refer to the "Fill in Gaps" and "Backbone" sections as needed.
The Fill in Gaps section covers the knowledge acquired in a strong high school mathematics education. Refer to it as needed, or begin there to develop your core skills.
The Backbone section supports the ideas in the Royal Road. Refer to it to strengthen your understanding of the ideas in the main track and to take those ideas further.
The greatest hurdles are motivation and coming to understand the language of mathematics.
See also this list of videos.
Fill in Gaps
Royal Road to Differential Geometry and Physics
Sets for Mathematics
Categorical approach to set theory by F. William Lawvere.
Backbone reference:
Mechanics
Classical mechanics of physics by Lev Landau.
Prerequisite:
Backbone reference:
Tensor Analysis on Manifolds
Tensor analysis by Richard Bishop and Samuel Goldberg.
Prerequisite:
Backbone reference:
Lectures on Differential Geometry
Differential geometry by Shlomo Sternberg.
Prerequisite:
Backbone reference:
Cohomology & Differential Forms
Cohomology and differential forms by Isu Vaisman.
Backbone reference:
Backbone
Topology: A Categorical Approach
Mathematical analysis by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
Applications of Lie Groups to Differential Equations
Applications of Lie Groups to Differential Equations by Peter Olver.