Webpage of the module 661955 T1 2024/25
Welcome to the module Differential Geometry 661955 for the BSc in Mathematics at the University of Hull, academic year 2024/25. In this module we study curves, surfaces and general topology.
If you have any questions please feel free to either
All the module information will be posted on this page, as well as on Canvas. The links to the reference material are:
Each week we have:
Times and venues are as follows:
Lecture 1: Thursday 16:00-18:00 in Wilberforce Building - Lecture Room 5
Lecture 2: Friday 16:00-18:00 in Wilberforce Building - Lecture Room 16
Tutorial: Time and venues as follows
Please check on MyTimetable every week to see if there are any changes to the following session times and rooms.
This module will be assessed as follows:
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 module Lecture Notes are available here. Topics covered in each lecture are as follows:
| Class # | Date | Time | Topics |
|---|---|---|---|
| 1 | 26/09/24 | 16:00 - 18:00 | Canvas, Assessment etc. Level, parametrized, smooth curves. Tangent vector. Rectifiabile curves. |
| 2 | 27/09/24 | 09:00 - 10:00 | Formula: Length of smooth curves. Length of circle and portion of Helix. |
| 3 | 27/09/24 | 16:00 - 18:00 | Arc-Length. Scalar product. Geometric and algebraic definitions. Bilinearity, Symmetry, Differentiation of SP. Speed. Reparametrizations. Regular and singular points. |
| 4 | 03/10/24 | 16:00 - 18:00 | Regularity and unit-speed reparametrization. Arc-length as unit speed reparametrization. Closed curves. Existsence of period. Examples. |
| 5 | 04/10/24 | 09:00 - 10:00 | Curvature: motivation. Curvature for unit speed curves. Example: Circle. Curvature for regular curves. Hyperbolic functions. Example: curvature of Catenary. |
| 6 | 04/10/24 | 16:00 - 18:00 | Vector Product in $\mathbb{R}^3$: Algebraic definition and geometric properties. Formula: curvature of regular curves (no proof). Examples. Hints for Homework 1. |
| 7 | 10/10/24 | 15:00 - 16:00 | Notation: Arc-length reparametrization, curvature, catenary example. Example: Plane curves. Frenet-Frame. Frenet-Frame and reparametrizations. Helix example. |
| 8 | 10/10/24 | 16:00 - 18:00 | Torsion for unit speed curves. Torsion, general formula. Example of calculation of curvature and torsion. Exercises of Homework 1. |
| 9 | 11/10/24 | 16:00 - 18:00 | Example: Rotated circle. Summary: Calculations on curves. Frenet-Serret equations. FTSC (no proof). Torsion of planar curves. |
| 10 | 17/10/24 | 16:00 - 18:00 | Curves with Constant curvature and zero torsion. Proof: Curvature and Torsion Formulas. Proof of FTSC. Topology: Definition. Trivial, discrete, euclidean topologies. |
| 11 | 18/10/24 | 09:00 - 10:00 | Balls are open in $\mathbb{R}^n$. Closed sets. Topology through closed sets. Zariski topology. Comparing topologies. Cofinite topology. |
| 12 | 18/10/24 | 16:00 - 18:00 | Convergence. Metric spaces. Topology induced by the metric. Interior, closure, boundary in general topological spaces. |
| 13 | 24/10/24 | 15:00 - 16:00 | Limit points are contained in the closure. The converse is false: co-countable topology example. Characterization of interior and closure in metric space. Density. |
| 14 | 24/10/24 | 16:00 - 18:00 | Hausdorff spaces. Metric spaces are Hausdorff. Metrizable spaces. Metrizable spaces are Hausdorff. Continuity. Example in $\mathbb{R}^m$. Exercises of Homework 2. |
| 15 | 25/10/24 | 16:00 - 18:00 | Continuity. Sequential continuity. Subspace topology. Topological basis. Topology induced by basis. Product topology. Connectedness. Examples. |
| 16 | 31/10/24 | 16:00 - 18:00 | Intervals. Intermediate value Theorem. Path-connectedness. Examples. Topologist Curve. Surfaces. Topology in $\mathbb{R}^n$. Smooth functions. Differential. Jacobian. |
| 17 | 01/11/24 | 09:00 - 10:00 | Diffeomorphism and local diffeomorphism. Inverse function Theorem. Examples. Surfaces: Definition. Examples of the plane. |
| 18 | 01/11/24 | 16:00 - 18:00 | Examples: Cylinder, Graphs. Double Cone is not a surface. Regular charts and regular surfaces. Examples: Plane, Cylinder, Graphs, Sphere. Spherical coordinates. |
| 19 | 07/11/24 | 16:00 - 18:00 | |
| 20 | 08/11/24 | 09:00 - 10:00 | Example: non-regular surface. Reparametrizations. Transition maps. Functions between surfaces. Smooth functions. Unit normal. Orientability. Orientable surfaces. |
| 21 | 08/11/24 | 16:00 - 18:00 | Exercises of Homework 3. Exercises of Homework 4. Exercises of Homework 5. |
| 22 | 14/11/24 | 16:00 - 18:00 | |
| 23 | 15/11/24 | 09:00 - 10:00 | |
| 24 | 15/11/24 | 16:00 - 17:00 | |
| 25 | 21/11/24 | 16:00 - 18:00 | Diffeomorphisms between surfaces. Tangent space. Examples. Differential of smooth functions. Differential of smooth functions in coordinates. |
| 26 | 22/11/24 | 09:00 - 10:00 | Properties of differential of smooth functions. Linear Algebra preliminaries. Examples of surfaces: Level Surfaces, Quadrics, Ruled Surfaces, Surfaces of Revolution. |
| 27 | 22/11/24 | 16:00 - 17:00 | First fundamental form (FFF): Abstract definition and expression in coordinates. Length of curves on surfaces. |
| 28 | 28/11/24 | 16:00 - 18:00 | Local isometries. Local isometries preserve length of curves and FFF. Angles on surfaces. Angles between curves. Conformal maps. |
| 29 | 29/11/24 | 09:00 - 10:00 | Conformal maps and FFF. Conformal parametrizations. Conformally flat surfaces. |
| 30 | 29/11/24 | 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 | 05/12/24 | 16:00 - 18:00 | 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 | 06/12/24 | 09:00 - 10:00 | Normal and Geodesic curvatures. Euler’s Theorem. Local shape of surface: Elliptic, Hyperbolic, Parabolic and Planar points. Local Structure Theorem. |
| 33 | 06/12/24 | 16:00 - 17:00 | Umbilical points and structure theorem at umbilics. |
| 34 | 12/12/24 | 16:00 - 18:00 | Revision and Exam Preparation. |
| 35 | 13/12/24 | 09:00 - 10:00 | Revision and Exam Preparation. |
| 36 | 13/12/24 | 16:00 - 17:00 | Revision and Exam Preparation. |
There will be 5 Homework papers in total:
Each homework paper:
| Due date | Homework # | Topics |
|---|---|---|
| 08/10/24 | 1 | Curve length, regularity, reparametrization. |
| 22/10/24 | 2 | Curvature, torsion, Frenet Frame. |
| 05/11/24 | 3 | Frenet-Serret. Topological spaces: Convergence, Interior, Closure. Topology of Metric Spaces. |
| 19/11/24 | 4 | Topological spaces. Density. Connectedness. Regular surfaces. Tangent space. First Fundamental Form. |
| 03/12/24 | 5 | First and Second Fundamental Forms. Gauss and Weingarten maps. Curvatures. Local shape. |
Homework papers submitted outside of Canvas or more than 24 hours after the Due Date will NOT BE MARKED
Please submit PDFs only. Either: