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This list of books provides the most direct and rigorous route to understanding differential geometry. See the image on the right for a visual treatment of its dependencies.
Each selection thoroughly addresses its topics.
There are other books for more specific topics. These are the core.
The greatest hurdles are motivation and coming to understand the language of mathematics.
See also this list of videos.
Fill in Gaps
Basic Mathematics
Review of arithmetic, algebra, trigonometry, logic, and geometry by Serge Lang.
Calculus
Overview of Calculus by Tom Apostol.
Royal Road to Differential Geometry and Physics
Sets for Methematics
Categorical approach to set theory by F. William Lawvere.
Backbone reference:
Linear Algebra
Linear algebra by Georgi Shilov.
Mechanics
Physics by Lev Landau.
The Classical Theory of Fields
Physics by Lev Landau.
Tensor Analysis on Manifolds
Tensor analysis by Richard Bishop and Samuel Goldberg.
Lectures on Differential Geometry
Differential geometry by Shlomo Sternberg.
Cohomology & Differential Forms
Cohomology and differential forms by Isu Vaisman.
Backbone
Ordinary Differential Equations
Ordinary differential equations by Vladimir Arnold.
Set Theory and Metric Spaces
Set theory and metric spaces by Irving Kaplansky.
Foundations of Analysis
Analysis, intro to numbers, by Edmund Landau.
Principles of Mathematical Analysis
Mathematical analysis by Walter Rudin.
Topology: A Categorical Approach
Mathematical analysis by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
Complex Analysis
Complex analysis by Lars Ahlfors.
Applications of Lie Groups to Differential Equations
Applications of Lie Groups to Differential Equations by Peter Olver.
Algebra Chapter 0
Algebra by Paolo Aluffi. Easier than Lang's, but less direct.
Algebra
Algebra by Serge Lang. The most direct approach to the subject.