Spring 2026
Guangbo Xu
Course Description:
This is a continuation of the course 532. The topics include
1. De Rham Cohomology and Hodge Theorem
2. Vector Bundles, Connections, and Characteristic Classes
3. Complex Manifolds (Riemann Surfaces) and Holomorphic Vector Bundles
4. Basic Gauge Theory (Yang-Mills Theory and Narasimhan-Seshadri Theorem)
Depending on the students' demand, the topics can be changed during the semester.
Textbook:
References: My Lecture Notes, together with Kobayashi-Nozimu: Foundations of Differential Geometry and Warner: Foundations of Differentiable Manifolds and Lie Groups
Prerequisite:
Manifold theory, basic topology, and preliminary knowledge about partial differential equations
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Spring 2025
Guangbo Xu
Course Description:
This is a continuation of the course 532: Introduction to Differential Geometry. However it differs from 533 in previous semesters. Depending on time and progress, topics may include:
1. Review connections and curvatures on vector bundles.
2. Connections on principal bundles.
3. Harmonic forms and Hodge Theorem.
4. Chern classes.
5. Basic gauge theory.
References:
Kobayashi-Nozimu: Foundations of Differential Geometry
Jost: Riemannian Geometry and Geometric Analysis
Warner: Foundations of Differentiable Manifolds and Lie Groups
Textbook:
Kobayashi-Nozimu: Foundations of Differential Geometry
Prerequisite:
532
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Fall 2024
Daniel Ketover
Course Description:
Introduction to Riemannian geometry
Textbook:
Introduction to Smooth Manifolds (Lee), Riemannian geometry (Do Carmo)
Prerequisite:
point set topology
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Spring 2024
Li Chi
Course Description:
This is a continuation of the course 532: Introduction to Differential Geometry. Depending on time and progress, topics may include:
1. Jacobi Fields and conjugate points
2. Isometric Immersions
3. Hopf-Rinow, Hadamard Theorems
4. Variations of energy, Bonnet-Myers theorem
5. Rauch Comparison Theorem
6. Harmonic forms, Hodge Theorem
7. Holonomy groups
References:
do Carmo: Riemannian Geometry
Jost: Riemannian Geometry and Geometric Analysis
Besse: Einstein manifolds
Textbook:
do Carmo, Riemannian Geometry
Prerequisite:
point set topology, multivariable calculus, 532: introduction to differential geometry
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Spring 2020
Jian Song
Course Description:
Differential geometry is the study of geometric properties of curves, surfaces, and their higher dimensional analogues using the methods of calculus. It has a long and rich history, and, in addition to its intrinsic mathematical value and important connections with various other branches of mathematics, it has many applications in various physical sciences. In this course, we will study differential manifolds, Riemannian metrics, Levi-Civita connections, curvature tensors, geodesics and space forms and possibly comparison theorems in Riemannian geometry.
Textbook:
My own notes
Prerequisite:
Real Analysis
Schedule of Sections: