# MATHS 735 : Analysis on Manifolds and Differential Geometry

## Science

### Course Prescription

Studies surfaces and their generalisations, smooth manifolds, and the interaction between geometry, analysis and topology; it is a central tool in many areas of mathematics, physics and engineering. Topics include Stokes' theorem on manifolds and the celebrated Gauss Bonnet theorem. Strongly recommended: MATHS 333 and 340.

### Course Overview

This is a core graduate course that is recommended for and pure mathematics graduate student and also for many students taking applied mathematics, physics, or engineering that may be interested in applying the ideas of continuously varying mathematical structures to problems in science.

It explores and develops  the interaction between geometry, analysis,  and topology and is extremely useful for those wishing to obtain a conceptual and practical understanding of many of the tools used in other areas of mathematics such as complex analysis, ordinary partial differential equations, partial differential equations as well as the applications of these things to, for example inverse problems and dynamics. There are also strong links with algebraic geometry.

### Course Requirements

Prerequisite: MATHS 332

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement Capability 5: Independence and Integrity

### Learning Outcomes

By the end of this course, students will be able to:
1. Understand that the subject combines ideas from analysis and geometry and that the geometric aspect imparts some rigidity to the structure (Capability 1 and 2)
2. Understand how local analysis and geometry impacts on global structure (Capability 1, 2, 3, 4 and 5)
3. Know how to design and execute proofs of theorems in this area (Capability 1, 2, 3, 4 and 5)
4. know apply the knowledge to other areas such as complex analysis and theoretical physics (Capability 1, 2 and 4)
5. Have a working knowledge of the basic tools of differential geometry, including differential manifolds, forms fields and tensors (Capability 1, 3 and 4)
6. Have knowledge of how to combine tools and techniques from many different areas of mathematics in order to prove new results and obtain a heightened level of understanding (Capability 1, 2, 3 and 4)

### Assessments

Assessment Type Percentage Classification
Assignments 40% Individual Coursework
Final Exam 60% Individual Coursework
1 2 3 4 5 6
Assignments
Final Exam

### Key Topics

Topics:

Analytic Euclidean  geometry
Curves - some general theory and tools
The local theory of plane curves and space curves
Plane curves - global theory
The local theory of immersed surfaces
The intrinsic geometry of surfaces
Riemannian geometry (- especially in two dimensions)
The global geometry of surfaces

### Special Requirements

No special requirements

This course is a standard [15] point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect [36] hours of lectures,  [44] hours of reading and thinking about the content and [70] hours of work on assignments and/or test preparation.

### Delivery Mode

#### Campus Experience

Attendance is expected at scheduled activities.
The course will not include live online events.
Attendance on campus is required for the exam.
The activities for the course are scheduled as a standard weekly timetable.

### Learning Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

Complete lecture notes will be provided.

Klingenberg, Wilhelm, A course in differential geometry.

O’Neill, Barrett, Elementary differential geometry.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Class Representatives

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page .

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter .

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course students may be asked to submit coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 02/11/2021 09:55 p.m.