## MATH4DG Differential Geometry

First Semester |

Since the time of the ancient Greeks, mathematicians and philosophers have been interested in the geometry of curves and surfaces, for example the Euclidean plane and the surface of the earth. From the end of the 19th century onwards with the work of Riemann, however, a powerful mathematical theory of much more general classes of curved spaces arose; this is what is nowadays understood as differential geometry.

In this module we cover:

- The concept of a smooth manifold
- The tangent space and tangent vector fields
- The Levi-Civita connection, parallel transport, geodesics and curvature
- Semi-Riemannian geometry (with particular emphasis on the Riemannian and Lorentzian cases)

Differential geometry has many applications in modern mathematics (both pure and applied) and theoretical physics (in particular general relativity and the standard model of particle physics). This module is therefore aimed to all students with interests in pure and applied mathematics, and in physics. Students who wish to do the module MATH4GR "Introduction to General Relativity" in the second half of semester 1, must do MATH4DG in the first half of the semester first.

### Prerequisites

MATH 202, MATH 203

### Main literature:

Lecture notes from the resource webpage and lecture notes by Prof. Christian Bär (University of Potsdam, Germany).

### Additional (but not compulsory) literature:

- Andrew Pressley, ebook
*Elementary Differential Geometry*(available for free download on the library webpage) - John M. Lee,
*Introduction to Smooth Manifolds*, Springer - Michael Spivak,
*Comprehensive introduction to differential geometry* - John M. Lee,
*Riemannian manifolds : an introduction to curvature*, Springer - Isaac Chavel,
*Riemannian Geometry*, A Modern Introduction - Robert Wald,
*General Relativity* - Sean M. Carroll,
*Spacetime and geometry : an introduction to general relativity*, Addison Wesley, 2004 (available in the science library, or alternatively, see the online lecture notes)

Feel free to browse the science library for further books yourself.

### Lecturers

Florian Beyer (phone 479-7768, email: fbeyer@maths.otago.ac.nz, room 218)

### Lecture times

3 Lectures per week. 1 Tutorial every two weeks. The times (and other information) is posted on the resource webpage.

NOTICE: This module starts in week 2 of the semester and finished at the end of week 7!

### Internal Assessment

3 written assignments

### Final exam

3 hours. Date to be decided as early as possible during the semester.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = (2E + A)/3**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.