08-17-2026, 04:27 PM
Differential Geometry: Connections, Curvature, and Characteristic Classes
Author: Loring W. Tu
Publication date: 2017
Publisher: Springer
Series: Graduate Texts in Mathematics, Vol. 275
Loring W. Tu’s Differential Geometry is a graduate-level introduction that develops the subject around two fundamental ideas: connections and curvature. Rather than presenting differential geometry simply as a collection of computations involving curves and surfaces, Tu gradually builds toward the modern geometric framework of vector bundles, principal bundles, differential forms, and characteristic classes. The exposition also follows the historical development of the subject, passing through landmarks such as Gauss’s Theorema Egregium, the curvature tensor, geodesics, and the Gauss–Bonnet theorem.
The book begins with curvature and vector fields and then reformulates curvature using differential forms. It proceeds to geodesics and the Gauss–Bonnet theorem before introducing the algebraic and topological machinery required for the second half. Vector bundles, connections, curvature forms, Pontryagin classes, Euler classes and Chern classes then lead naturally to principal bundles. The ultimate objective is an explanation of Chern–Weil theory, showing how geometric information encoded by curvature produces topological invariants—one of the central bridges between differential geometry and algebraic topology.
This is therefore considerably more advanced than a traditional first course centered on curves and surfaces in $\mathbb{R}^3$. Tu assumes some familiarity with manifolds; after the first chapter the reader needs differential forms, while de Rham cohomology becomes necessary for roughly the final third. Tu specifically points readers toward his earlier An Introduction to Manifolds for this background. Exercises are integrated throughout, and selected hints and solutions appear at the end. For someone interested in the relationship between geometry, topology and mathematical physics, the book provides a particularly attractive route from basic curvature to the sophisticated theory of characteristic classes.
Key takeaways
Springer — Differential Geometry by Loring W. Tu
Author: Loring W. Tu
Publication date: 2017
Publisher: Springer
Series: Graduate Texts in Mathematics, Vol. 275
Loring W. Tu’s Differential Geometry is a graduate-level introduction that develops the subject around two fundamental ideas: connections and curvature. Rather than presenting differential geometry simply as a collection of computations involving curves and surfaces, Tu gradually builds toward the modern geometric framework of vector bundles, principal bundles, differential forms, and characteristic classes. The exposition also follows the historical development of the subject, passing through landmarks such as Gauss’s Theorema Egregium, the curvature tensor, geodesics, and the Gauss–Bonnet theorem.
The book begins with curvature and vector fields and then reformulates curvature using differential forms. It proceeds to geodesics and the Gauss–Bonnet theorem before introducing the algebraic and topological machinery required for the second half. Vector bundles, connections, curvature forms, Pontryagin classes, Euler classes and Chern classes then lead naturally to principal bundles. The ultimate objective is an explanation of Chern–Weil theory, showing how geometric information encoded by curvature produces topological invariants—one of the central bridges between differential geometry and algebraic topology.
This is therefore considerably more advanced than a traditional first course centered on curves and surfaces in $\mathbb{R}^3$. Tu assumes some familiarity with manifolds; after the first chapter the reader needs differential forms, while de Rham cohomology becomes necessary for roughly the final third. Tu specifically points readers toward his earlier An Introduction to Manifolds for this background. Exercises are integrated throughout, and selected hints and solutions appear at the end. For someone interested in the relationship between geometry, topology and mathematical physics, the book provides a particularly attractive route from basic curvature to the sophisticated theory of characteristic classes.
Key takeaways
- Modern rather than purely classical: the emphasis is on manifolds, connections, curvature and bundles rather than only curves and surfaces.
- Clear destination: much of the book systematically prepares the reader for Chern–Weil theory and characteristic classes.
- Strong geometry–topology connection: it explains how local differential-geometric quantities such as curvature can encode global topological information.
- Not really a beginner's first geometry book: some knowledge of manifolds, differential forms and eventually de Rham cohomology is important.
Springer — Differential Geometry by Loring W. Tu
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