Table of contents

Differential Geometry from First Principles

A beginner-friendly path from curves and surfaces to manifolds, curvature, and geometric thinking

Read each section in order. Every title can be opened as a TheoryTrace document.

• Cover
• Copyright
• How to read this book
• Introduction
• Chapter 1: What Differential Geometry Studies
• Chapter 2: The Essential Linear Algebra
• Chapter 3: The Essential Multivariable Calculus
• Chapter 4: Curves in Euclidean Space
• Chapter 5: Curvature and Torsion of Curves
• Chapter 6: Surfaces as Parametrized Shapes
• Chapter 7: The First Fundamental Form
• Chapter 8: The Gauss Map and the Second Fundamental Form
• Chapter 9: Gaussian and Mean Curvature
• Chapter 10: Geodesics and Shortest Paths
• Chapter 11: Intrinsic Geometry and Gauss’s Theorema Egregium
• Chapter 12: Covariant Derivatives and Parallel Transport on Surfaces
• Chapter 13: The Gauss–Bonnet Theorem
• Chapter 14: From Surfaces to Manifolds
• Chapter 15: Tangent Spaces and Smooth Maps
• Chapter 16: Vector Fields, Flows, and Lie Brackets
• Chapter 17: Differential Forms and Integration
• Chapter 18: Stokes’s Theorem as a Unifying Principle
• Chapter 19: Riemannian Metrics
• Chapter 20: Connections, Geodesics, and Curvature on Manifolds
• Chapter 21: Ricci Curvature, Scalar Curvature, and Geometric Meaning
• Chapter 22: Classical Examples and Computations
• Chapter 23: How Differential Geometry Connects to Modern Mathematics
• Chapter 24: A Guided Problem-Solving Workshop
• Conclusion