Outline

Overview

This unit is an introduction to Differential Geometry, one of the core pillars of modern mathematics. Using ideas from calculus of several variables, we develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. For students, this provides the first taste of the investigation on the deep relation between geometry and topology of mathematical objects, highlighted in the classic Gauss-Bonnet Theorem. Differential geometry also plays an important part in both classical and modern theoretical physics. The unit aims to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to remind the students about all the content covered in the mathematical units for previous years, most importantly the key ideas in vector calculus, along with some applications. It also helps to prepare the students for honours courses like Riemannian Geometry. By doing this unit you will further appreciate the beauty of mathematics which originated from the need to solve practical problems, develop skills in understanding the geometry of the surrounding environment, prepare yourself for future study or the workplace by developing advanced critical thinking skills and gain a deep understanding of the underlying rules of the Universe.

Unit details and rules

Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prerequisites ?	(A mark of 65 or greater in 12cp of MATH2XXX units of study) or [12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3078 or MATH3961 or MATH3962 or MATH3963 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3978 or MATH3979)]
Corequisites ?	None
Prohibitions ?	MATH3968
Assumed knowledge ?	Vector calculus, differential equations and real analysis, for example MATH2X21 and MATH2X23
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Emma Carberry, emma.carberry@sydney.edu.au
Lecturer(s)	Emma Carberry, emma.carberry@sydney.edu.au
Tutor(s)	Emma Carberry, emma.carberry@sydney.edu.au

Assessment

The census date for this unit availability is 2 September 2024

Details
Criteria

Type	Description	Weight	Due	Length
Supervised exam ?	Final exam Written exam, including computational and proof-based questions.	60%	Formal exam period	2 hours
Outcomes assessed:
Presentation	Tutorial presentation Group presentation of a tutorial problem (with individual assessment)	5%	Multiple weeks	Not more than 10 minutes
Outcomes assessed:
Assignment	Assignment 1, group component Written assignment	5%	Week 06 Due date: 08 Sep 2024 at 23:59 Closing date: 18 Sep 2024	2 weeks
Outcomes assessed:
Assignment	Assignment 1, individual component Written assignment	10%	Week 06 Due date: 08 Sep 2024 at 23:59 Closing date: 18 Sep 2024	2 weeks
Outcomes assessed:
Assignment	Assignment 2 Written assignment	15%	Week 12 Due date: 27 Oct 2024 at 23:59 Closing date: 06 Nov 2024	2 weeks
Outcomes assessed:
Small continuous assessment	Weekly tutorial problems Write up of a tutorial problem, only the best 9 submissions count.	5%	Weekly	Typically 1 question, may have subparts.
Outcomes assessed:
= group assignment ?

Assessment summary

Detailed information for each assessment can be found on Canvas.

Final exam: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator

Assessment criteria

The University awards common result grades, set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name	Mark range	Description
High distinction	85 - 100	Representing complete or close to complete mastery of the material.
Distinction	75 - 84	Representing excellence, but substantially less than complete mastery.
Credit	65 - 74	Representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence.
Pass	50 - 64	Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
Fail	0 - 49	When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see guide to grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

Deduction of 5% of the maximum mark for each calendar day after the due date.
After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Late submissions for weekly tutorial problems are not permitted. Note that only the best 9 of 12 tutorial problems count towards your grade.

Academic integrity

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

Use of generative artificial intelligence (AI) and automated writing tools

You may only use generative AI and automated writing tools in assessment tasks if you are permitted to by your unit coordinator. If you do use these tools, you must acknowledge this in your work, either in a footnote or an acknowledgement section. The assessment instructions or unit outline will give guidance of the types of tools that are permitted and how the tools should be used.

Your final submitted work must be your own, original work. You must acknowledge any use of generative AI tools that have been used in the assessment, and any material that forms part of your submission must be appropriately referenced. For guidance on how to acknowledge the use of AI, please refer to the AI in Education Canvas site.

The unapproved use of these tools or unacknowledged use will be considered a breach of the Academic Integrity Policy and penalties may apply.

Studiosity is permitted unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission as detailed on the Learning Hub’s Canvas page.

Outside assessment tasks, generative AI tools may be used to support your learning. The AI in Education Canvas site contains a number of productive ways that students are using AI to improve their learning.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Support for students

The Support for Students Policy 2023 reflects the University’s commitment to supporting students in their academic journey and making the University safe for students. It is important that you read and understand this policy so that you are familiar with the range of support services available to you and understand how to engage with them.

The University uses email as its primary source of communication with students who need support under the Support for Students Policy 2023. Make sure you check your University email regularly and respond to any communications received from the University.

Learning resources and detailed information about weekly assessment and learning activities can be accessed via Canvas. It is essential that you visit your unit of study Canvas site to ensure you are up to date with all of your tasks.

If you are having difficulties completing your studies, or are feeling unsure about your progress, we are here to help. You can access the support services offered by the University at any time:

Support and Services (including health and wellbeing services, financial support and learning support)
Course planning and administration
Meet with an Academic Adviser

Weekly schedule

WK	Topic	Learning activity
Week 01	1. Parameterised curves; 2. Frenet equations; 3. The fundamental theorem	Lecture and tutorial (4 hr)
Week 02	1. The theorem of turning tangents; 2. Basic topology; 3. The implicit function theorem	Lecture and tutorial (4 hr)
Week 03	1. Regular surfaces; 2. More on the implicit function theorem; 3. Tangent planes; 4. The first fundamental form	Lecture and tutorial (4 hr)
Week 04	1. Area; 2. Orientation; 3. The Gauss map	Lecture and tutorial (4 hr)
Week 05	1. The second fundamental form; 2. Principal curvatures; 3. Gauss; 4. Mean curvatures	Lecture and tutorial (4 hr)
Week 06	1. Gauss curvature continued; 2. Minimal surfaces	Lecture and tutorial (4 hr)
Week 07	1. Conformal maps; 2. Gauss’ theorema egregium; 3. The Gauss-Mainardi-Codazzi equations	Lecture and tutorial (4 hr)
Week 08	1. Covariant derivative; 2. Parallel transport; 3. Geodesics; 4. Geodesic curvature	Lecture and tutorial (4 hr)
Week 09	1. The exponential map; 2. Geodesics as length-minimising curves	Lecture and tutorial (4 hr)
Week 10	1. Theorem of turning tangents; 2. Gauss-Bonnet theorem	Lecture and tutorial (4 hr)
Week 11	1. Poincaré-Hopf index theorem; 2. Morse’s theorem	Lecture and tutorial (4 hr)
Week 12	Abstract surfaces/manifolds	Lecture and tutorial (4 hr)

Attendance and class requirements

Attendance at tutorials is expected.

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Required readings

All readings for this unit can be accessed on the Library eReserve link available on Canvas.

Required text:

Differential Geometry of Curves and Surfaces, 2nd edition, by Manfredo do Carmo, available as an e-book through the library website or on order at the student co-op.

For a more introductory text you may wish to refer to:

Elementary Differential Geometry, by Andrew Crossley.

For the latter part of the course, also useful is:

Differential Forms and Applications, by Manfredo do Carmo.

Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University's graduate qualities and are assessed as part of the curriculum.

Outcomes
Graduate qualities

At the completion of this unit, you should be able to:

LO1. demonstrate knowledge and understanding of fundamental concepts and theorems in differential geometry.
LO2. apply fundamental theorems and concepts of differential geometry in order to solve geometric problems
LO3. understand and apply geometric concepts to analyse examples to draw conclusions
LO4. evaluate geometric quantities such as torsion and curvature
LO5. Synthesise knowledge across a range of topics and write novel mathematical proofs.

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1	Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.
GQ2	Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.
GQ3	Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.
GQ4	Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.
GQ5	Inventiveness Generating novel ideas and solutions.
GQ6	Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.
GQ7	Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.
GQ8	Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.
GQ9	Influence Engaging others in a process, idea or vision.

Outcome map

Learning outcomes	Graduate qualities
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

Quiz removed in order to reduce volume of assessment. More recorded lecture content in response to student feedback.

Additional information

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non compliance have increased. Everyone has a responsibility for health and safety at work. The Universitys Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Attendance and class requirements

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

MATH4068: Differential Geometry

Semester 2, 2024 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Support for students

Weekly schedule

Weekly schedule

Attendance and class requirements

Study commitment

Required readings

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Work, health and safety

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect