📌 Course Information

🕒 Class Schedule & Venues

👥 Teaching Staff & Contact

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Recent Announcement

• (7 Sept) Quota of the course has been added to 80, and an extra tutorial session is offered. For those who are on the waiting list, please swap to T1B and the system will get you enrolled in the course. </aside>

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Remark about Prerequisites

Workable knowledge and geometric understanding of Multivariable Calculus, and basic computational skills in Linear Algebra are essential for the survival of the course.

For aiming at better grades (say B or above), you need to be familiar with abstract concepts in Linear Algebra such as vector spaces, basis and dimensions, linear transformations, dual spaces, quadratic form, eigenvalues and eigenvectors, etc.

Take the Background Self-Test to check your background.

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Office Hours

You are welcome to make an appointment with Prof. Fong to discuss about the course. Please book an available time slot by clicking the button below:

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📖 Course Description

Differential Geometry is the study of geometric shapes, curves, and curved surfaces using the techniques of multivariable calculus and linear algebra. This course provides a rigorous, classical introduction to the differential geometry of curves and surfaces in Euclidean spaces ($\mathbb{R}^n$), especially when for $n=3$, serving as a crucial foundation for modern Riemannian geometry, general relativity, and geometric analysis.

The main topics covered include:

• Curve Theory: Regular curves, arc length parameterization, curvature $\kappa$, torsion $\tau$, the Frenet-Serret apparatus and equations, and the Fundamental Theorem of Space Curves.
• Surface Theory: Regular surfaces in $\mathbb{R}^3$, regular hypersurfaces in $\mathbb{R}^{n+1}$, tangent planes, the Gauss map, the Weingarten operator (shape operator), first and second fundamental forms, principal curvatures, Gaussian curvature $K$, mean curvature $H$, ruled surfaces, and minimal surfaces.
• Intrinsic Geometry: Isometries, Christoffel symbols, covariant derivatives Gauss's Theorema Egregium, Gauss-Codazzi’s equations, parallel transport, geodesics, first and second variations of arc-length.
• Gauss-Bonnet’s Theorem: Geodesic curvature, differential forms, Cartan’s structure equations, the local and global Gauss-Bonnet Theorems, Chern’s instrinsic proof of Gauss-Bonnet Theorem for surfaces.

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Please read the AI usage policy, and the homework submission guideline in the [syllabus]

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Course Materials

Course Calendar / Blackboard Slides / Lecture Videos