This list of books provides the most direct and rigorous route to understanding differential geometry, the mathematical language of physics. 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/or read several together to acquire the necessary language and understand the motivations for each idea. The greatest hurdles are the motivation to learn and developing an understanding of the language of mathematics.
See the image on the right for a visual representation of its dependencies.
Also see this list of video lectures.
A further set of texts extending this one, but working with the same basics has been produced leading all the way up and through gauge field theory, quantum mechanics, algebraic geometry, and quantum field theory here.
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.
Fill in Gaps
Royal Road to Differential Geometry and Physics
Sets for Mathematics
The Classical Theory of Fields
Physics by Lev Landau.
Tensor Analysis on Manifolds
Lectures on Differential Geometry
Topology: A Categorical Approach
Topology by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla. Click here for the Open Access version.
Applications of Lie Groups to Differential Equations
Applications of Lie Groups to Differential Equations by Peter Olver.