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This list of books provides the most direct and rigorous route to understanding differential geometry.

Its selections are made because reasons.

The greatest hurdles are motivation and coming to understand the language of mathematics.

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: Set Theory and Metric Spaces by Kaplansky, Foundations of Analysis by Edmund Landau.

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

Complex analysis by Paolo Aluffi.

Algebra

Algebra by Serge Lang.