Differential Geometry (MATH3405)

Course overview

Curves: Parameterised curves, regular curves, arc length. Local theory, Frenet frame. Global properties of plane curves.
Regular surfaces: Change of parameterisation, differential functions on surfaces. The tangent plane, the differential of a map. First fundamental form, area functional.
Geometry of the Gauss map: fundamental properties, local coordinates. Vector fields. Minimal surfaces and other applications.
Intrinsic and extrinsic geometry of surfaces: isometries, conformal maps.
Selected topics from: Theorema egregium. Parallel transport, geodesics. Gauss-Bonnet Theorem and applications. Exponential map. Geodesic polar coordinates.

The geometry of curves and surfaces in three dimensional space is studied using techniques from linear algebra and calculus. We will determine various geometric quantities associated to a curve or surface. An example of such a quantity is the length of the acceleration vector at a point of a curve in R3 (where the curve is parameterised by unit speed). This length is the curvature of the curve at that point. We will show that a non-constant curve with curvature zero at every point must be a line. This is another key example of how local behavior (the curvature at each point is zero) determines global behavior (the curve is a straight line).

Similar to the relationship between local and global properties, we will also compare intrinsic and extrinsic properties of curves and surfaces. For example, bending a sheet of paper changes its extrinsic geometry, but not its intrinsic geometry. An example of an intrinsic quantity (that is not changed by bending the sheet of paper) is the Gauss curvature, which is introduced in the last third of this course. The definition of the Gauss curvature makes essential use of the position of a surface in space. However, Gauss showed that it is nevertheless independent of the position in space and only depends on the geometry of the surface. He was so pleased with this result that he called it "Theorema Egregium," a remarkable theorem.

Another concept introduced in this course is the topology, or shape, of a geometrical object. Two surfaces have the same topology if you can deform one into another by bending, twisting, stretching and deforming, but not tearing it. Two spheres of different radii certainly have the same topology, but we will see that they are distinguished by their curvatures. The same is true for any sphere and a (non-trivial) ellipsoid. Towards the end of the course, we will present the Gauss-Bonnet Theorem, relating the total curvature of a closed surface to its topology.

In the last two weeks we will briefly study higher-dimensional generalisations of curves and surfaces:ﾠdifferentiable manifolds. These are extremely important objects, present in a great number of areas within modern Mathematics and Physics.ﾠ

Course requirements

Assumed background

The prerequisite for this course isﾠﾠMATH2001, and I will assume knowledge of the concepts and methods introduced there (and in its prerequisites MATH1051 and MATH1052). There are two recommended prerequisites, MATH2400 (or MATH2401) and MATH2301. They are recommended not so much for content, but rather for general mathematical training.ﾠ

If you wish to revise the background material, it should suffice to review the notes, homework, assignments and exams of the prerequisite. If you feel enthusiastic, an excellent way of revising and deepening this material would be to work through Michael Spivak's "Calculus on Manifolds," which starts with the inner product on Rn and ends with Green's Theorem and the Divergence Theorem.

Prerequisites

You'll need to complete the following courses before enrolling in this one:

MATH2000 or MATH2001

Recommended prerequisites

We recommend completing the following courses before enrolling in this one:

MATH2400 or (MATH2401), MATH2301

Incompatible

You can't enrol in this course if you've already completed the following:

MATH7435 (co-taught).

Course contact

Course staff

Timetable

The timetable for this course is available on the UQ Public Timetable.

Additional timetable information

All classes will be conducted on campus. Consult your personal timetable for times and locations. Students are expected to attend these sessions in person unless they have a valid reason for being unable to attend (such as illness).

Important: if you are ill, then do not attend any classes in person. Alternative arrangements can be organised – consult Blackboard for details. 

Tutorial commence week 2.

Aims and outcomes

This course aims to give a hands-on introduction to differential geometry, in particular to

1. introduce important notions such as curvature through the theory of curves and surfaces in 3-space, and
2. motivate and introduce more abstract notions such as that of a manifold.

Learning outcomes

After successfully completing this course you should be able to:

Assessment

Assessment summary

A hurdle is an assessment requirement that must be satisfied in order to receive a specific grade for the course. Check the assessment details for more information about hurdle requirements.

Assessment details

Assignments

Mode

Written

Category

Tutorial/ Problem Set

Weight

50%

Due date

Assignment 1: 9/08/2024 5:00 pm

Assignment 2: 23/08/2024 5:00 pm

Assignment 3: 6/09/2024 5:00 pm

Assignment 4: 20/09/2024 5:00 pm

Assignment 5: 11/10/2024 5:00 pm

Assignment 6: 25/10/2024 5:00 pm

Learning outcomes

L01, L02, L03, L04, L05, L06, L08

Task description

Submission guidelines

Assignments will be submitted electronically in Blackboard. Submissions in LaTEX are strongly encouraged.

Deferral or extension

You may be able to apply for an extension.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Late submission

A penalty of 10% of the maximum possible mark will be deducted per 24 hours from time submission is due for up to 7 days. After 7 days, you will receive a mark of 0.

You are required to submit assessable items on time. If you fail to meet the submission deadline for any assessment item then the listed penalty will be deducted per day for up to 7 calendar days, at which point any submission will not receive any marks unless an extension has been approved. Each 24-hour block is recorded from the time the submission is due.

Final exam

• Hurdle

Mode

Written

Category

Examination

Weight

50%

Due date

End of Semester Exam Period

2/11/2024 - 16/11/2024

Learning outcomes

L01, L02, L03, L04, L05, L06, L07, L08

Task description

Hurdle requirements

Exam details

Submission guidelines

Deferral or extension

You may be able to defer this exam.

See ADDITIONAL ASSESSMENT INFORMATION for the extension and deferred examination information relating to this assessment item.

Course grading

Full criteria for each grade is available in the Assessment Procedure.

Additional course grading information

Note from the above criteria: students will need a mark of at least 40% on the final exam to achieve a passing grade in this course, regardless of their other marks; and a mark of at least 35% on the final exam to achieve a grade of 3 in this course, regardless of their other marks.

Supplementary assessment

Supplementary assessment is available for this course.

Should you fail a course with a grade of 3, you may be eligible for supplementary assessment. Refer to my.UQ for information on supplementary assessment and how to apply. 

Supplementary assessment provides an additional opportunity to demonstrate you have achieved all the required learning outcomes for a course.  

If you apply and are granted supplementary assessment, the type of supplementary assessment set will consider which learning outcome(s) have not been met.  

Supplementary assessment in this course will be a 2-hour examination similar in style to the end-of-semester examination. To receive a passing grade of 3S4, you must obtain a mark of 50% or more on the supplementary assessment. 

Additional assessment information

Artificial Intelligence

The assessment tasks in this course evaluate students’ abilities, skills and knowledge without the aid of Artificial Intelligence (AI). Students are advised that the use of AI technologies to develop responses is strictly prohibited and may constitute misconduct under the Student Code of Conduct.

Applications for Extensions to Assessment Due Dates

Extension requests are submitted online via my.UQ – applying for an extension. Extension requests received in any other way will not be approved. Additional details associated with extension requests, including acceptable and unacceptable reasons, may be found at my.UQ.

Please note:

• Requests for an extension to an assessment due date must be submitted through your my.UQ portal and you must provide documentation of your circumstances, as soon as it becomes evident that an extension is needed. Your application must be submitted on or before the assessment item's due date and time.
• Applications for extension can take time to be processed so you should continue to work on your assessment item while awaiting a decision. We recommend that you submit any completed work by the due date, and this will be marked if your application is not approved. Should your application be approved, then you will be able to resubmit by the agreed revised due date.
• If an extension is approved, you will be notified via your my.UQ portal and the new date and time for submission provided. It is important that you check the revised date as it may differ from the date that you requested.
• If the basis of the application is a medical condition, applications should be accompanied by a medical certificate dated prior to the assignment due date. If you are unable to provide documentation to support your application by the due date and time you must still submit your application on time and attach a written statement (Word document) outlining why you cannot provide the documentation. You must then upload the documentation to the portal within 24 hours.
• If an extension is being sought on the basis of exceptional circumstances, it must be accompanied by supporting documentation (eg. Statutory declaration).
• For extensions based on a SAP you may be granted a maximum of 7 days (if no earlier maximum date applies). See the Extension or Deferral availability section of each assessment for details. Your SAP is all that is required as documentation to support your application. However, additional extension requests for the assessment item will require the submission of additional supporting documentation e.g., a medical certificate. All extension requests must be received by the assessment due date and time.
• Students may be asked to submit evidence of work completed to date. Lack of adequate progress on your assessment item may result in an extension being denied.
• If you have been ill or unable to attend class for more than 14 days, you are advised to carefully consider whether you are capable of successfully completing your courses this semester. You might be eligible to withdraw without academic penalty - seek advice from the Faculty that administers your program.
• There are no provisions for exemption from an assessment item within UQ rules. If you are unable to submit an assessment piece then, under special circumstances, you may be granted an exemption, but may be required to submit alternative assessment to ensure all learning outcomes are met.

Applications to defer an exam

In certain circumstances you can apply to take a deferred examination for in-semester and end-of-semester exams. You'll need to demonstrate through supporting documentation how unavoidable circumstances prevented you from sitting your exam. If you can’t, you can apply for a one-off discretionary deferred exam.

Deferred Exam requests are submitted online via mySi-net. Requests received in any other way will not be approved. Additional details associated with deferred examinations, including acceptable and unacceptable reasons may be found at my.UQ.

Please note:

• Applications can be submitted no later than 5 calendar days after the date of the original exam.
• There are no provisions to defer a deferred exam. You need to be available to sit your deferred examination.
• Your deferred examination request(s) must have a status of "submitted" in mySI-net to be assessed.
• All applications for deferred in-semester examinations are assessed by the relevant school. Applications for deferred end-of-semester examinations are assessed by the Academic Services Division.
• You’ll receive an email to your student email account when the status of your application is updated.
• If you have a medical condition, mental health condition or disability and require alternative arrangements for your deferred exam you’ll need to complete the online alternative exam arrangements through my.UQ. This is in addition to your deferred examinations request. You need to submit this request on the same day as your request for a deferred exam or supplementary assessment. Contact Student Services if you need assistance completing your alternative exam arrangements request.

Learning resources

You'll need the following resources to successfully complete the course. We've indicated below if you need a personal copy of the reading materials or your own item.

Library resources

Find the required and recommended resources for this course on the UQ Library website.

Learning activities

The learning activities for this course are outlined below. Learn more about the learning outcomes that apply to this course.

Policies and procedures

University policies and procedures apply to all aspects of student life. As a UQ student, you must comply with University-wide and program-specific requirements, including the:

• Student Code of Conduct Policy
• Student Integrity and Misconduct Policy and Procedure
• Assessment Procedure
• Examinations Procedure
• Reasonable Adjustments - Students Policy and Procedure

Learn more about UQ policies on my.UQ and the Policy and Procedure Library.