Differential Geometry (MAST90143)
Graduate courseworkPoints: 12.5Not available in 2025
About this subject
Overview
| Fees | Look up fees |
|---|
This subject extends notions from calculus, linear algebra and differential equations to study spaces with geometric structures. The concepts introduced are of great importance in mathematics, physics, and all areas in which local properties of spaces are used to model systems.
Topics include: smooth manifolds, vector bundles, multilinear algebra; Frobenius’ theorem, exterior differentiation, Lie differentiation, flows of vector fields; connections and curvature; bilinear forms, metrics, length, volume, Levi-Civita connection; parallel transport, geodesics, holonomy; connections on principal bundles; examples including Lie groups, hyperbolic geometry and homogeneous spaces.
Additional topics may include: second fundamental form and minimal submanifolds; Jacobi fields and applications to topology; constant curvature and Einstein metrics; Hodge star operator, Hodge Laplacian and harmonic forms; Lorentzian geometry and Einstein's equations; Kähler geometry; symplectic geometry; gauge theory.
Intended learning outcomes
After completing this subject, student should be able to:
- demonstrate understanding of the basic notions of Differential Geometry, including smooth manifolds, vector bundles, Riemannian metrics, connections and curvature;
- functionally use connections, curvature and geodesics;
- perform geometric calculations in local coordinates;
- explain and apply major foundational results in differential geometry;
- demonstrate knowledge of important examples of Lie groups and homogeneous spaces; and
- pursue further studies in differential geometry and related areas.
Generic skills
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- Problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- Analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- Collaborative skills: the ability to work in a team; and
- Time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 4 March 2025
Eligibility and requirements
Prerequisites
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST30026 | Metric and Hilbert Spaces | Semester 2 (On Campus - Parkville) |
12.5 |
AND
One of
| Code | Name | Teaching period | Credit Points |
|---|---|---|---|
| MAST20009 | Vector Calculus |
Semester 2 (On Campus - Parkville)
Semester 1 (On Campus - Parkville)
|
12.5 |
| MAST20032 | Vector Calculus: Advanced | Semester 1 (On Campus - Parkville) |
12.5 |
Corequisites
None
Non-allowed subjects
None
Inherent requirements (core participation requirements)
The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.
Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home
Last updated: 4 March 2025
Assessment
| Description | Timing | Percentage |
|---|---|---|
Written assignment
| Early in the teaching period | 20% |
Written Assignment
| Second half of the teaching period | 20% |
Written Assignment
| Late in the teaching period | 20% |
Final exam
| During the examination period | 40% |
Last updated: 4 March 2025
Dates & times
Not available in 2025
What do these dates mean
Visit this webpage to find out about these key dates, including how they impact on:
- Your tuition fees, academic transcript and statements.
- And for Commonwealth Supported students, your:
- Student Learning Entitlement. This applies to all students enrolled in a Commonwealth Supported Place (CSP).
Subjects withdrawn after the census date (including up to the ‘last day to withdraw without fail’) count toward the Student Learning Entitlement.
Last updated: 4 March 2025
Further information
- Texts
- Available to Study Abroad and/or Study Exchange Students
Last updated: 4 March 2025