Math230A: Differential Geometry

Differential geometry is a branch of mathematics that studies geometric structures on manifolds and is fundamental to understanding Floer homology. This course introduces basic concepts such as manifolds, tangent bundles, Riemannian metrics, and curvature, laying the foundation for subsequent study of topology and geometry.

Course Information

Course Name: Differential Geometry

Course Number: Math230A

Instructor: Fan Ye

Course Content:

This course covers the following main topics:

• Basic concepts of manifolds
• Tangent bundles and cotangent bundles
• Riemannian metrics
• Connections and curvature
• Geodesics
• Submanifold theory

Textbooks:

Main References:

• Differential Geometry: Bundles, Connections, Metrics and Curvature, Chapters 1-16, Clifford H. Taubes

Other suggested references:

• An Introduction to Differentiable Manifolds and Riemannian Geometry, William M. Boothby
• Foundations of Differentiable Manifolds and Lie Groups, Frank W. Warner
• Differential Forms in Algebraic Topology, Raoul Bott, Loring W. Tu
• Vector Bundles & K-Theory, Allen Hatcher

Tentative Course Outline: Notes will be posted on the course page after the classes.

• Weeks 1-2: Smooth manifolds and Lie groups [Taubes, Chapters 1-2]
• Weeks 3-4: Vector bundles [Taubes, Chapters 3-6]
• Weeks 5-6: Metrics and geodesics [Taubes, Chapters 7-9]
• Weeks 7-8: de Rham cohomology and covariant derivative [Taubes, Chapter 12]
• Weeks 9-10: Levi-Civita connections and Riemann curvature tensors [Taubes, Chapters 15, 14.1]
• Weeks 11-13: Characteristic classes and principal bundles [Taubes, Chapters 14, 10-11]
• Week 14: Yang-Mills equation and applications

Prerequisites: An undergraduate-level understanding of manifolds. An undergraduate may look at the first chapter of Cliff’s book for preparation.

Course History

Note: This course information is compiled based on related course content from Harvard University.