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This list of books provides the most direct and rigorous route to understanding differential geometry.
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.
![Landau Course in Theoretical Physics V1 Cover.jpg](/images/thumb/6/6f/Landau_Course_in_Theoretical_Physics_V1_Cover.jpg/300px-Landau_Course_in_Theoretical_Physics_V1_Cover.jpg)
Mechanics
Physics by Lev Landau.
![Landau Course in Theoretical Physics V2 Cover.jpg](/images/thumb/b/b2/Landau_Course_in_Theoretical_Physics_V2_Cover.jpg/300px-Landau_Course_in_Theoretical_Physics_V2_Cover.jpg)
The Classical Theory of Fields
Physics by Lev Landau.
![Bishop Tensor Analysis Cover.jpg](/images/thumb/9/92/Bishop_Tensor_Analysis_Cover.jpg/300px-Bishop_Tensor_Analysis_Cover.jpg)
Tensor Analysis on Manifolds
Tensor analysis by Richard Bishop and Samuel Goldberg.
![Sternberg Differential Geometry Cover.jpg](/images/thumb/5/54/Sternberg_Differential_Geometry_Cover.jpg/300px-Sternberg_Differential_Geometry_Cover.jpg)
Lectures on Differential Geometry
Differential geometry by Shlomo Sternberg.
![Vaisman Cohomology and Differential Forms Cover.jpg](/images/thumb/e/ed/Vaisman_Cohomology_and_Differential_Forms_Cover.jpg/300px-Vaisman_Cohomology_and_Differential_Forms_Cover.jpg)
Cohomology & Differential Forms
Cohomology and differential forms by Isu Vaisman.
Backbone
![Arnold Ordinary Differential Equations Cover.jpg](/images/thumb/e/eb/Arnold_Ordinary_Differential_Equations_Cover.jpg/300px-Arnold_Ordinary_Differential_Equations_Cover.jpg)
Ordinary Differential Equations
Ordinary differential equations by Vladimir Arnold.
![Kaplansky Set Theory and Metric Spaces Cover.jpg](/images/thumb/b/b7/Kaplansky_Set_Theory_and_Metric_Spaces_Cover.jpg/300px-Kaplansky_Set_Theory_and_Metric_Spaces_Cover.jpg)
Set Theory and Metric Spaces
Set theory and metric spaces by Irving Kaplansky.
![E Landau Foundations of Analysis Cover.jpg](/images/thumb/5/5c/E_Landau_Foundations_of_Analysis_Cover.jpg/300px-E_Landau_Foundations_of_Analysis_Cover.jpg)
Foundations of Analysis
Analysis, intro to numbers, by Edmund Landau.
![Rudin Principles of Mathematical Analysis Cover.jpg](/images/thumb/d/d6/Rudin_Principles_of_Mathematical_Analysis_Cover.jpg/300px-Rudin_Principles_of_Mathematical_Analysis_Cover.jpg)
Principles of Mathematical Analysis
Mathematical analysis by Walter Rudin.
![Bradley Bryson Terrilla Topology A Categorical Appoach Cover.jpg](/images/f/ff/Bradley_Bryson_Terrilla_Topology_A_Categorical_Appoach_Cover.jpg)
Topology: A Categorical Approach
Mathematical analysis by Tai-Danae Bradley, Tyler Bryson, Josn Terrilla.
![Ahlfors Complex Analysis Cover.jpg](/images/thumb/d/df/Ahlfors_Complex_Analysis_Cover.jpg/300px-Ahlfors_Complex_Analysis_Cover.jpg)
Complex Analysis
Complex analysis by Lars Ahlfors.
![Olver Applications of Lie Groups to Differential Equations Cover.jpg](/images/thumb/e/e4/Olver_Applications_of_Lie_Groups_to_Differential_Equations_Cover.jpg/300px-Olver_Applications_of_Lie_Groups_to_Differential_Equations_Cover.jpg)
Applications of Lie Groups to Differential Equations
Applications of Lie Groups to Differential Equations by Peter Olver.
![Aluffi Algebra Chapter 0 Cover.jpg](/images/thumb/6/67/Aluffi_Algebra_Chapter_0_Cover.jpg/300px-Aluffi_Algebra_Chapter_0_Cover.jpg)
Algebra Chapter 0
Complex analysis by Paolo Aluffi.
![Lang Algebra Cover.jpg](/images/4/43/Lang_Algebra_Cover.jpg)
Algebra
Algebra by Serge Lang.