Read
This list of books provides the most direct and rigorous route to understanding differential geometry. Each selection thoroughly addresses its subject matter. 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 for each idea.
See the image on the right for a visual treatment of its dependencies.
The Royal Road to Differential Geometry and Physics is the list's core. While on that track, you should refer to the "Fill in Gaps" and "Backbone" sections as needed or as you desire.
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
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 Mathematics
Categorical approach to set theory by F. William Lawvere.
Backbone reference:
Linear Algebra
Overview of linear algebra by Georgi Shilov.
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
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.
Ordinary Differential Equations
Ordinary differential equations by Vladimir Arnold.
Topology: A Categorical Approach
Topology 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.