Overview
Basic properties and foundational results of differential geometry and semi-Riemannian geometry, manifolds, tangent vectors and vector fields, covariant derivatives, geodesics and curvature.
Basic properties and foundational results of differential geometry and semi-Riemannian geometry, manifolds, tangent vectors and vector fields, covariant derivatives, geodesics and curvature.
About this paper
| Paper title | Differential Geometry |
|---|---|
| Subject | Mathematics |
| EFTS | 0.0833 |
| Points | 10 points |
| Teaching period | Semester 1 (On campus) |
| Domestic Tuition Fees ( NZD ) | $620.00 |
| International Tuition Fees | Tuition Fees for international students are elsewhere on this website. |
- Restriction
- MATH 401-412
- Limited to
- BA(Hons), BSc(Hons), PGDipArts, PGDipSci, MA (Thesis), MSc, MAppSc, PGDipAppSc, PGCertAppSc
- Contact
Mathematics 400-level programme coordinator: Dr Fabien Montiel fabien.montiel@otago.ac.nz
- Teaching staff
- Paper Structure
Main topics (19 lectures, 50 min each):
Lectures 1-5:
- Charts.
- C-infinity-atlas.
- Differentiable structures.
- Differentiable manifolds
- Basic examples.
Lectures 5-9:
- C-infinity functions.
- C-infinity curves.
- Tangent vectors.
- Tangent vector spaces.
- Derivations.
- Directional derivatives.
- Coordinate bases.
Lecture 9:
- Co(tangent)-vectors.
- Co(tangent)-vector spaces.
- Dual bases.
Lectures 9-10:
- Semi-Riemannian forms.
- Orthonormal bases.
- Riemannian and Lorentzian forms.
Lectures 10-12:
- (Co)tangent bundle.
- (Co)tangent vector fields.
- Semi-Riemannian (Riemannian, Lorentzian) metrics.
- Semi-Riemannian (Riemannian, Lorentzian) manifolds.
Lectures 12-14:
- Levi-Civita connection.
- Levi-Civita Theorem.
Lectures 15-17:
- Vector fields along curves.
- Velocity field.
- Covariant derivatives.
- Parallel transport.
- Geodesic equations examples.
Lectures 18-19:
- Introduction to curvature.
- Teaching Arrangements
19 lectures (50 minutes each).
Tutorials: Weekly drop-in sessions for help with assignments.
Assessment (40% of total mark): 3 written assignments.
Exam (60% of total mark)
- Graduate Attributes Emphasised
Critical Thinking, Interdisciplinary Perspective, Lifelong Learning.
View more information about Otago's graduate attributes.- Learning Outcomes
On completion of the study of this paper, students are expected to:
- Understand fundamental approaches in differential geometry.
- Know important properties of manifolds, covariant derivatives and curvature.
- Understand how to rigorously proof theorems in differential geometry.
- Understand the power of working with geometric concepts by means of examples.