This course introduces students to the key concepts and techniques of Differential Geometry. Possible topics include:

• Surfaces in Euclidean space
• General differentiable manifolds
• Tangent spaces and vector fields
• Differential forms
• Riemannian manifolds
• Gauss-Bonnet theorem

Note: Graduate students attend joint classes with undergraduates but are assessed separately.

Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

On satisfying the requirements of this course, students will have the knowledge and skills to:

1. Explain the concepts and language of differential geometry and its role in modern mathematics
2. Analyse and solve complex problems using appropriate techniques from differential geometry
3. Apply problem-solving with differential geometry to diverse situations in physics, engineering or other mathematical contexts
4. Apply differential geometry techniques to specific research problems in mathematics or other fields

Recommended Resources

Recommended student system requirements 

ANU courses commonly use a number of online resources and activities including:

• video material, similar to YouTube, for lectures and other instruction
• two-way video conferencing for interactive learning
• email and other messaging tools for communication
• interactive web apps for formative and collaborative activities
• print and photo/scan for handwritten work
• home-based assessment.

To fully participate in ANU learning, students need:

• A computer or laptop. Mobile devices may work well but in some situations a computer/laptop may be more appropriate.
• Webcam
• Speakers and a microphone (e.g. headset)
• Reliable, stable internet connection. Broadband recommended. If using a mobile network or wi-fi then check performance is adequate.
• Suitable location with minimal interruptions and adequate privacy for classes and assessments.
• Printing, and photo/scanning equipment

For more information please see https://www.anu.edu.au/students/systems/recommended-student-system-requirements

Staff Feedback

Students will be given feedback in the following forms in this course:

• Feedback through student representatives.
• Feedbacks on their assignments and exams.

Student Feedback

ANU is committed to the demonstration of educational excellence and regularly seeks feedback from students. Students are encouraged to offer feedback directly to their Course Convener or through their College and Course representatives (if applicable). The feedback given in these surveys is anonymous and provides the Colleges, University Education Committee and Academic Board with opportunities to recognise excellent teaching, and opportunities for improvement. The Surveys and Evaluation website provides more information on student surveys at ANU and reports on the feedback provided on ANU courses.

Class Schedule

Week/Session	Summary of Activities	Assessment
1	Definition of manifolds Immersion, embeddings of submanifolds
2	Manifold construction theorem Definition of functions and maps on manifolds
3	Tangent/cotangent spaces The tangent bundle and its manifold structure
4	Vector bundles Transition functions. Examples: Tangent bundles
5	Whitney embedding theorem Applications of partitions of unity
6	Lie brackets The flow of a vector field
7	Lie derivative Frobenius theorem and applications
8	Connection 1-forms Connections and covariant derivatives
9	Riemannian metrics Levi-Civita connections
10	Gauss Lemma Exponential normal coordinates
11	Tensor bundles Curvature of a connection
12	Gauss’s Theorema Egregium Ricci curvature and scalar curvature

Assessment Summary

Assessment task	Value	Due Date	Return of assessment	Learning Outcomes
Written assignment 1	15 %	16/03/2021	27/03/2021	1,2,3,4
Written assignment 2	15 %	06/04/2021	17/04/2021	1,2,3,4
Written assignment 3	15 %	11/05/2021	22/05/2021	1,2,3,4
Assignment 4	15 %	28/05/2021	05/06/2021	1,2,3,4
Lecture/Workshop Zoom participation	10 %	22/02/2021	28/05/2021	1,2,3,4
Take home final exam	10 %	03/06/2021	01/07/2021	1,2,3,4
Report	20 %	18/06/2021	01/07/2021	1,2,3,4

* If the Due Date and Return of Assessment date are blank, see the Assessment Tab for specific Assessment Task details

Policies

ANU has educational policies, procedures and guidelines, which are designed to ensure that staff and students are aware of the University’s academic standards, and implement them. Students are expected to have read the Academic Misconduct Rule before the commencement of their course. Other key policies and guidelines include:

• Student Assessment (Coursework) Policy and Procedure
• Special Assessment Consideration Policy and General Information 
• Student Surveys and Evaluations
• Deferred Examinations
• Student Complaint Resolution Policy and Procedure

Assessment Requirements

The ANU is using Turnitin to enhance student citation and referencing techniques, and to assess assignment submissions as a component of the University's approach to managing Academic Integrity. For additional information regarding Turnitin please visit the Academic Integrity . In rare cases where online submission using Turnitin software is not technically possible; or where not using Turnitin software has been justified by the Course Convener and approved by the Associate Dean (Education) on the basis of the teaching model being employed; students shall submit assessment online via ‘Wattle’ outside of Turnitin, or failing that in hard copy, or through a combination of submission methods as approved by the Associate Dean (Education). The submission method is detailed below.

Moderation of Assessment

Marks that are allocated during Semester are to be considered provisional until formalised by the College examiners meeting at the end of each Semester. If appropriate, some moderation of marks might be applied prior to final results being released.

Participation

We believe that discussing mathematics is one of the best ways to master the material. Students are expected to engage actively and respectfully in cooperative problem-solving during the workshops and laboratories. Students are strongly encouraged to attend lecturer, office hours and interactive sessions and ask questions over the Zoom!

Examination(s)

Please note, that where a date range is used in the Assessment Summary in relation to exams, the due date and return date for end of semester exams indicate the approximate timeframe in which the exam will be held and the date official end of Semester results are released on ISIS. Students should consult the course wattle site for the details of the exam.

Assessment Task 1

Written assignment 1

A written assignment involving problem-solving, proofs of theorems and extension of theory (15%).

Assessment Task 2

Written assignment 2

A written assignment involving problem-solving, proofs of theorems and extension of theory (15%).

Assessment Task 3

Written assignment 3

A written assignment involving problem-solving, proofs of theorems and extension of theory (15%)

Assessment Task 4

Assignment 4

A written assignment involving problem-solving, proofs of theorems and extension of theory (15%).

Assessment Task 5

Lecture/Workshop Zoom participation

The date range for this task comprises the start of the semester and the end of the teaching period. Further details can be found on the Course Wattle site.

Students are required to log in via their ANU Zoom accounts and activate their video to attend lectures and workshops, unless prior arrangement has been made with the lecturers. Weekly Zoom workshops led by demonstrators offer students a chance to work cooperatively on problems related to the class material.

Attending lectures and workshops is one of the most important class activities, as they represent one of the best opportunities to learn the material. Worksheets will be posted on Wattle by the Friday of the week preceding the workshop. Students work through worksheet problems and write up the solutions during the workshops, with guidance from the demonstrator. The workshop worksheet is not handed in as an assignment; solutions are written in a workbook provided by you but the workbook is not submitted. Note that we will provide written solutions to workshop problems by the Friday of the week of the workshop.

Assessment Task 6

Take home final exam

The final exam paper and details will be available on Wattle.

Assessment Task 7

Report

For those postgraduate students, 20% for reading some research articles relevant to the course and writing a (up-to 10 pages) report (Due by 11:59 pm June 20). See Wattle course information for details.

Academic Integrity

Academic integrity is a core part of the ANU culture as a community of scholars. At its heart, academic integrity is about behaving ethically, committing to honest and responsible scholarly practice and upholding these values with respect and fairness.

The ANU commits to assisting all members of our community to understand how to engage in academic work in ways that are consistent with, and actively support academic integrity. The ANU expects staff and students to be familiar with the academic integrity principle and Academic Misconduct Rule, uphold high standards of academic integrity and act ethically and honestly, to ensure the quality and value of the qualification that you will graduate with.

The Academic Misconduct Rule is in place to promote academic integrity and manage academic misconduct. Very minor breaches of the academic integrity principle may result in a reduction of marks of up to 10% of the total marks available for the assessment. The ANU offers a number of online and in person services to assist students with their assignments, examinations, and other learning activities. Visit the Academic Skills website for more information about academic integrity, your responsibilities and for assistance with your assignments, writing skills and study.

Online Submission

You will be required to electronically sign a declaration as part of the submission of your assignment. Please keep a copy of the assignment for your records. Unless an exemption has been approved by the Associate Dean (Education) submission must be through Turnitin.

Hardcopy Submission

For some forms of assessment (hand written assignments, art works, laboratory notes, etc.) hard copy submission is appropriate when approved by the Associate Dean (Education). Hard copy submissions must utilise the Assignment Cover Sheet. Please keep a copy of tasks completed for your records.

Late Submission

Late submission permitted. Late submission of assessment tasks without an extension are penalised at the rate of 5% of the possible marks available per working day or part thereof. Late submission of assessment tasks is not accepted after 10 working days after the due date, or on or after the date specified in the course outline for the return of the assessment item. Late submission is not accepted for take-home examinations.

Referencing Requirements

Accepted academic practice for referencing sources that you use in presentations can be found via the links on the Wattle site, under the file named “ANU and College Policies, Program Information, Student Support Services and Assessment”. Alternatively, you can seek help through the Students Learning Development website.

Returning Assignments

Each student is responsible for checking that their marks are entered correctly on Wattle; written records of the marks should be saved until they have been confirmed on Wattle. Any discrepancies should be reported a timely manner and no later than the end of Week 12.

Extensions and Penalties

Extensions and late submission of assessment pieces are covered by the Student Assessment (Coursework) Policy and Procedure. Extensions may be granted for assessment pieces that are not examinations or take-home examinations. If you need an extension, you must request an extension in writing on or before the due date. If you have documented and appropriate medical evidence that demonstrates you were not able to request an extension on or before the due date, you may be able to request it after the due date.

Privacy Notice

The ANU has made a number of third party, online, databases available for students to use. Use of each online database is conditional on student end users first agreeing to the database licensor’s terms of service and/or privacy policy. Students should read these carefully. In some cases student end users will be required to register an account with the database licensor and submit personal information, including their: first name; last name; ANU email address; and other information.
In cases where student end users are asked to submit ‘content’ to a database, such as an assignment or short answers, the database licensor may only use the student’s ‘content’ in accordance with the terms of service – including any (copyright) licence the student grants to the database licensor. Any personal information or content a student submits may be stored by the licensor, potentially offshore, and will be used to process the database service in accordance with the licensors terms of service and/or privacy policy.
If any student chooses not to agree to the database licensor’s terms of service or privacy policy, the student will not be able to access and use the database. In these circumstances students should contact their lecturer to enquire about alternative arrangements that are available.

Distribution of grades policy

Academic Quality Assurance Committee monitors the performance of students, including attrition, further study and employment rates and grade distribution, and College reports on quality assurance processes for assessment activities, including alignment with national and international disciplinary and interdisciplinary standards, as well as qualification type learning outcomes.

Since first semester 1994, ANU uses a grading scale for all courses. This grading scale is used by all academic areas of the University.

Support for students

The University offers students support through several different services. You may contact the services listed below directly or seek advice from your Course Convener, Student Administrators, or your College and Course representatives (if applicable).

• ANU Health, safety & wellbeing for medical services, counselling, mental health and spiritual support
• ANU Diversity and inclusion for students with a disability or ongoing or chronic illness
• ANU Dean of Students for confidential, impartial advice and help to resolve problems between students and the academic or administrative areas of the University
• ANU Academic Skills and Learning Centre supports you make your own decisions about how you learn and manage your workload.
• ANU Counselling Centre promotes, supports and enhances mental health and wellbeing within the University student community.
• ANUSA supports and represents undergraduate and ANU College students
• PARSA supports and represents postgraduate and research students