Webpage of the Course 661955 T1 2023/24
Welcome to the Course of Differential Geometry 661955 for the BSc in Mathematics at the University of Hull, academic year 2023/24. This course studies curves and surfaces in 2D and 3D.
If you have any questions please feel free to email me. We will use the Tutorials to address homework and exam preparation. In addition, please do not hesitate to attend office hours.
All the course information will be posted on this page, as well as on Canvas. The links to the reference material are:
We have two Lectures and one Tutorial per week. The lectures are 2 hours long, and the tutorial is 1 hour long.
Pattern from Week 1 to Week 7:
Pattern from Week 8 to Week 12:
This course will be assessed as follows:
Each Homework paper is worth 100 points. The final Homework grade will be computed averaging the best 4.
Homework papers will be posted on Canvas.
Homework papers must be submitted on Canvas by 14:00 on the Due Date. The due dates are:
Homework papers submitted outside of Canvas or after the deadline will NOT BE MARKED
Please submit PDFs only. Either:
Curves in 2D and 3D: regular curves, curvature, Frenet frame.
Surfaces in 3D: regular surfaces, quadrics.
First Fundamental Form: maps preserving lengths, angles and areas.
Second Fundamental Form: Gaussian curvature, Gauss’s Theorem Egregium.
Topology: General topology, metric spaces, compactness, connectedness.
Lecture Notes: Available here
Differential Geometry Book: Pressley
Topology Book: Manetti
Background: For analysis please refer to Zorich
The course Lecture Notes are available here. Each week we have 2 Lectures of 2h each. Dates and topics are below.
| Lesson | Date | Time | Topics |
|---|---|---|---|
| 1 | 28/09/23 | 12:00 - 13:00 | Intro: Canvas page, Lecture Notes, Assignments, Assessment. Curves as Level sets. Parametrized curves. Examples. |
| 2 | 29/09/23 | 11:00 - 13:00 | Plotting curves in Python using Matplotlib and Plotly. Parametrized curves examples. Tangent vector. Definition of Length of $\gamma$ as limit of length of polygonals. |
| 3 | 29/09/23 | 16:00 - 18:00 | Proven that $L(\gamma)$ can be compute by integrating $| \dot \gamma |$ when $\gamma$ regular. Computed length of circle and portion of Helix. |
| 4 | 05/10/23 | 12:00 - 13:00 | Arc-Length. Examples. Scalar product in $\mathbb{R}^2$: geometric definition. SP in coordinates and in $\mathbb{R}^n$. Bilinearity, Symmetry, Differentiation of SP. |
| 5 | 06/10/23 | 11:00 - 13:00 | Speed. Reparametrizations. Regular and singular points. Unit speed reparametrization. Thm: Regularity is equivalent to existence of unit speed reparametrization. |
| 6 | 06/10/23 | 16:00 - 18:00 | Theorem: Arc-length as unit speed reparametrization. Periodic curves. Closed curves. Period of a closed curve. Theorem: Period of closed curve exists. Examples. |
| 7 | 12/10/23 | 12:00 - 13:00 | Curvature: motivation with Taylor formula. Curvature for unit speed curves. Curvature of circle of radius $R$ computed via reparametrization to unit speed curve. |
| 8 | 13/10/23 | 11:00 - 13:00 | Vector Product in $\mathbb{R}^3$: Algebraic definition and geometric properties. General formula for curvature of regular curves (no proof for now). Examples. |
| 9 | 13/10/23 | 16:00 - 18:00 | Plane curves. Signed curvature (SC). Geometric meaning of SC. SC characterizes plane curves (no proof). Correction of Homework 1. |
| 10 | 19/10/23 | 12:00 - 13:00 | Hyperbolic functions and their properties. Example of the catenary curve. |
| 11 | 20/10/23 | 11:00 - 13:00 | Torsion for unit speed curves. Torsion, general formula. Example of calculation of curvature and torsion. Frenet Frame. Frenet-Serret equations. |
| 12 | 20/10/23 | 16:00 - 18:00 | Frenet Frame of Helix. Characterization Theorem of 3D curves. New notations. Proof of general curvature formula. Geometric consequences of Frenet-Serret. |
| 13 | 26/10/23 | 12:00 - 13:00 | Definition of Topology. Trivial, discrete and euclidean topologies. Balls are open in $\mathbb{R}^n$. Closed sets. Topology through closed sets. Zariski topology. |
| 14 | 27/10/23 | 11:00 - 13:00 | Comparing topologies. Cofinite topology. Convergence. Metric spaces. Interior, closure, boundary in general topological spaces. |
| 15 | 27/10/23 | 16:00 - 18:00 | Limit points are contained in the closure. The converse is false: co-countable topology example. Characterization of interior and closure in metric space. Density. |
| 16 | 02/11/23 | 12:00 - 13:00 | Correction of Homework 2. Hausdorff space. Metric spaces are Hausdorff. Metrizable spaces. Metrizable spaces are Hausdorff. |
| 17 | 03/11/23 | 11:00 - 13:00 | Examples of Hausdorff spaces. Continuity. Subspace topology. Topological basis. Topology induced by basis. |
| 18 | 03/11/23 | 16:00 - 18:00 | Product topology. Connectedness. Examples. Connectedness is topological invariant. $\mathbb{R}^n$ is not homeomorphic to $\mathbb{R}$. |
| 19 | 09/11/23 | 12:00 - 13:00 | Sequential continuity: fixed one of the proofs. Characterization of connected sets in $\mathbb{R}$. Intermediate value theorem in topological space. |
| 20 | 10/11/23 | 11:00 - 13:00 | Path connectedness. Definition, Examples. Path connectedness implies connectedness. Converse is false: Topologist Curve. Surfaces: Introductory discussion. |
| 21 | 10/11/23 | 16:00 - 18:00 | Correction of Homework 3. Topology in $\mathbb{R}^n$. Smooth functions $f \colon \mathbb{R}^n \to \mathbb{R}^m$, differential, Jacobian. |
| 22 | 16/11/23 | 12:00 - 14:00 | Inverse function Theorem. Examples. Surface: Definition and Examples. Regular charts and regular surfaces. |
| 23 | 17/11/23 | 11:00 - 13:00 | Spherical coordinates. Level surfaces. Reparametrizations. |
| 24 | 17/11/23 | 16:00 - 17:00 | Transition maps. Functions between surfaces. Smooth functions. |
| 25 | 23/11/23 | 12:00 - 14:00 | Diffeomorphisms between surfaces. Tangent space and characterization. Examples. Differential of smooth functions. Differential of smooth functions in coordinates. |
| 26 | 24/11/23 | 11:00 - 13:00 | Properties of differential of smooth functions. Linear Algebra preliminaries. Examples of surfaces: Level Surfaces, Quadrics, Ruled Surfaces, Surfaces of Revolution. |
| 27 | 24/11/23 | 16:00 - 17:00 | First fundamental form (FFF): Abstract definition and expression in coordinates. Length of curves on surfaces. |
| 28 | 30/11/23 | 12:00 - 14:00 | Local isometries. Local isometries preserve length of curves and FFF. Angles on surfaces. Angles between curves. Conformal maps. |
| 29 | 01/11/23 | 11:00 - 13:00 | Conformal maps and FFF. Conformal parametrizations. Conformally flat surfaces. Unit normal. Orientability. Orientable surfaces. |
| 30 | 01/12/23 | 16:00 - 17:00 | Second Fundamental Form (SFF) of a chart. Examples. Gauss map and examples. Weingarten map. SFF as a bilinear form on tangent space. Equivalence with SFF of a chart. |
| 31 | 07/12/23 | 12:00 - 14:00 | Matrix of Weingarten map. Gaussian and mean curvatures $G$ and $H$. Formulas for $G$ and $H$. Principal curvatures and directions. Relationship to $G$ and $H$. Examples. |
| 32 | 08/12/23 | 11:00 - 13:00 | Normal and Geodesic curvatures. Euler’s Theorem. Local shape of surface: Elliptic, Hyperbolic, Parabolic and Planar points. Local Structure Theorem. |
| 33 | 08/12/23 | 16:00 - 17:00 | Umbilical points and structure theorem at umbilics. |
| 34 | 14/12/23 | 12:00 - 14:00 | Revision and Exam Preparation. |
| 35 | 15/12/23 | 11:00 - 13:00 | Revision and Exam Preparation. |
| 36 | 15/12/23 | 16:00 - 17:00 | Revision and Exam Preparation. |
Homework papers must be submitted on Canvas by 14:00 on the Due Date
Homework papers submitted outside of Canvas or after the deadline will NOT BE MARKED
Please submit PDFs only. Either:
| Due date | Homework Number | Topics |
|---|---|---|
| 11/10/23 | 1 | Curve length, regularity, reparametrization. |
| 25/10/23 | 2 | Curve length, regularity, reparametrization, curvature, torsion. |
| 08/11/23 | 3 | Curvature, torsion. Topological spaces: Convergence, Interior, Closure. Topology of Metric Spaces. |
| 22/11/23 | 4 | Metric Spaces. Topology induced by metric. Density. Connectedness. |
| 15/12/23 | 5 | First and Second Fundamental Forms. Gauss and Weingarten maps. Curvatures. Local shape. |