Spring 2027
Jian Song
Subtitle:
Course Description:
This course provides a focused introduction to modern Differential Geometry, emphasizing the role of curvature in the global analysis of Riemannian manifolds. We begin with a concise review of the foundations of Riemannian Geometry, including smooth manifolds, Riemannian metrics, the Levi-Civita connection, geodesics, and the basic notions of curvature (sectional, Ricci, and scalar curvature), with an emphasis on geometric intuition and key examples.The course then develops the central theme of Comparison Theorems in Riemannian Geometry, including distance and volume comparison results such as the Bishop–Gromov inequality, Myers’ theorem and Cheeger-Gromoll splitting theorem. These tools are used to relate curvature bounds to global geometric and topological properties, including diameter estimates, volume growth, and rigidity phenomena.
Textbook:
private lecture notes
Prerequisites:
real analysis and some basic knowledge of differential geometry
Schedule of Sections:
Previous Semesters:
- Spring 2026 - Xu, Guangbo
- Spring 2025 - Xu, Guangbo
- Spring 2024 - Li, Chi
- Spring 2020 Prof. Song