📌 Course Information

🕒 Class Schedule & Venues

👥 Teaching Staff & Contact

<aside> 🗣️

Recent Announcement

• (3 Nov - new!) Edward Witten will give a talk in HKUST IAS on 3pm - 4:30pm, 17 November (Tuesday). We will not have lecture of the course that day. I strongly recommend you to attend Witten’s talk. Here is the link: https://calendar.hkust.edu.hk/events/ias-distinguished-lecture-invitation-black-hole-thermodynamics
• (27 Oct) A new version of lecture notes — which I give it a musical name — “Sonate geometrique, op. 1” — is uploaded (old version still there). Major changes are:
  • Correction of many typos and careless mistakes over the whole set (thanks GPT-6)
  • Moving the Riemann curvature tensor to Chapter 3 (originally a small subsection in Chapter 4), and it is now standalone as Section 3.4
  • More examples and exercises added in Chapters 3 - 4. </aside>

<aside> 💯

Midterm

Date: 17 October 2026 (Saturday) Time: 2pm - 4pm Venue: CYT G010

Coverage: The exam will cover everything up to the lecture note 15 October 2026, but questions related to the content on 13 Oct and 15 Oct will be limited to 5% of points at most.

Allowed material: You can bring a piece of A4-size paper, with notes written on both sides. The only requirement is that the first fundamental form must be $dx^2 + dy^2$. There is no restriction on the second fundamental form.

No electronic device. No calculators.

</aside>

<aside> 📆

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:

</aside>

📖 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.

<aside> ⚠️

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

</aside>

Course Materials

<aside> 💡

You can find some interactive applets used in lecture by clicking the Lecture box in the calendar below.

</aside>

Course Calendar / Blackboard Slides / Lecture Videos