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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
Royal Road to Differential Geometry and Physics
Sets for Methematics
Categorical approach to set theory by F. William Lawvere. Backbone reference: Set Theory and Metric Spaces by Kaplansky, Foundations of Analysis by Edmund Landau.
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.