Programs
- Fundamental Algebra and Analysis I
- Fundamental Algebra and Analysis II
- Geometry and Topology
- Abstract Algebra I
- Abstract Algebra II
- Ordinary Differential Equations
- Measure Theory and Real Analysis
- Probability Theory
- Functions of One Complex Variable
- Functional Analysis
- Numerical Analysis
- Category Theory
- Number Theory
- Algebraic Topology
- Representation Theory
- Differential Geometry and Differential Manifolds
- Partial Differential Equations
- Stochastic Processes
- Mathematical Modeling
- Numerical Algebra and Optimization
- Numerical Simulation
- Regression and Generalized Linear Models
- Scheme Theory
- Sheaf Theory and Cohomology
- Riemannian Geometry
- Elliptic Curves
- Topics in Several Complex Variables
- Geometric Measure Theory
- Harmonic Analysis
- Analytic Theory of Biological Dynamics
Riemannian Geometry
Prerequisites:
Differential Geometry and Differential Manifolds
Course description:
"Riemannian Geometry" is a core branch of differential geometry, grounded in the groundbreaking ideas of the 19th-century mathematician Bernhard Riemann. It investigates the geometric properties of curved spaces (Riemannian manifolds) that possess arbitrary dimensions and complex structures. This course aims to systematically introduce the fundamental theories, essential tools, and applications of Riemannian geometry in modern mathematics and physics. It seeks to cultivate students' abilities to describe and analyze higher-dimensional spaces using geometric language, laying a solid foundation for further studies in cutting-edge fields such as differential topology, general relativity, and geometric analysis.
Learning objectives:
Master Fundamental Concepts: Develop a solid understanding of core concepts such as Riemannian manifolds, metric tensors, connections, geodesics, and curvature tensors, while building an intuitive geometric picture of curved spaces.
Utilize Core Tools: Become proficient in calculations involving covariant differentiation, various contractions of the Riemann curvature tensor (Ricci curvature, scalar curvature), and comprehend their geometric significance.
Understand Key Theorems: Grasp the statements and proof frameworks of central results like the Hopf-Rinow theorem, the classification of space forms, and comparison theorems (e.g., the Rauch comparison theorem, the Laplace comparison theorem, and volume comparison theorems).
Establish Connections and Applications: Apply the theories of Riemannian geometry to related fields, such as understanding the Einstein field equations in general relativity or laying the groundwork for advanced topics in geometric analysis like minimal submanifolds and the Ricci flow.
Cultivate Geometric Thinking: Train the ability to perform local calculations and analyze global properties on abstract manifolds, and develop the skill to describe complex structures of higher-dimensional spaces using geometric language
Detailed topics covered:
a. Basic notions: Riemannian metric, connection, curvature, etc
b. Classification of space forms
c. Variation theory for geodesics
d. Comparison geometry (Rauch comparison, Bishop-Gromov volume comparison, Bonnet-Myers-Cheng diameter comparion)
e. Advance topics: topological sphere theorem
References:
《Riemannian Geometry》 by do Carmo
《Riemannian Geometry》 by Peter Petersen