Fan Ye | Fall 2022 | Harvard University
Basic properties and examples of smooth manifolds, Lie groups, and vector bundles; Riemannian geometry (metrics, geodesics, Levi-Civita connections, and Riemann curvature tensors); principal bundles and associated vector bundles with their connections and characteristic classes.
General
References
Differential Geometry: Bundles, Connections, Metrics, and Curvature, Chapters 1-16 - Clifford H. Taubes
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 Geometry - Raoul Bott and Loring W. Tu
Vector Bundles and K-Theory - Allen Hatcher
Week-by-Week Lecture
- Class 1: Introduction to Manifolds (Cliff, Chap. 1)
- Class 2: Smooth Manifolds (Chap. 1)
- Class 3: Lie Group (Chap. 2)
- Class 4: Vector Bundle (Chap. 3)
- Class 5: Tangent Bundle
- Class 6: Tangent Bundle and Cotangent Bundle
- Class 7: Bundle Algebra and Bundle Maps (Chap. 4-5)
- Class 8 Metrics on Vector Bundles
- Class 9 Riemannian Metric
- Class 10: Geodesic (Chap. 8)
- Class 11: Geodesic Theorem (Chap. 9)
- Class 12: de Rham Cohomology (Chap. 12)
- Class 13: Exterior Derivative and Lie Derivative
- Class 14: Covariant Derivative (Chap. 11)
- Class 18: Covariantly Constant Section and Curvature
- Class 19: Bianchi Identity and Chern Class (Chap. 14)
- Class 20: Principal Bundle (Chap. 10)
- Class 21: Review
- Class 22: Connection on Principal Bundle (Chap. 11.4)
- Class 23: Connection and Covariant Derivative
- Class 24: Horizontal Lift and Yang-Mills Equation
- Class 25/26: Topological Invariants from Yang-Mills Equation