# ACM40090 Riemannian Differential Geometry

## Academic Year 2022/2023

This module introduces the concepts, due initially to Riemann, of n-dimensional surfaces 'manifolds' that appear to be Euclidean locally in a neighbourhood of each point but globally may have curvature and a concept of distance - think of the 2-dimensional example of the surface of the Earth. The concepts of manifolds and their Differential Geometry, vector and tensor fields defined on them, appear throughout modern Applied and Computational Mathematics from fluids to dynamical systems to Einstein's theory of General Relativity.

Topics covered will include:

[Tensors over a vector space]: Dual vector space, the dual basis, multilinear maps, the tensor product, tensor spaces of type (r,s), operations on tensors.

[Manifolds]: Definition of a manifold, charts, diffeomorphism.

[Vectors on a manifold]: Parameterized curves, directional derivatives as vectors, derivations, forms.

[Maps on tensors]: The pul-back and push-forward, diffeomorphism as coordinate transformations. One parameter families of diffeomorphisms and the Lie derivative.

[Connections and curvature]: Definition of a connection, transformation of the coordinate components of a connection. The covariant derivative. Parallel transport, geodesics. The torsion and curvature tensors. The Bianchi identities.

[Riemannian manifolds]: The metric tensor, the Riemann tensor, geodesics as extremal curves, geodesic deviation.

Show/hide contentOpenClose All

Curricular information is subject to change

Learning Outcomes:

1. Expalin the concept of a tensor over a vector space.
2. Describe the construction of the dual basis and its transformation properties.
3. Explain the concept of manifolds
4. Explain the constuction of vector and tensor fields on a manifold.
5. Describe the pull back and push forward of appropriate tensors.
6. Expalin the concept of a one-parameter family of diffeomorphisms and the Lie derivative.
7. Explain the concept of a connection and the corresponding differentiation of tensor fields.
8. Explain the concepts of parallel transport and curvature.
9. Calculate the equations of geodesic motion.
10. Compute the curvature of a manifold.

Student Effort Hours:
Student Effort Type Hours
Lectures

24

Tutorial

6

Specified Learning Activities

20

Autonomous Student Learning

50

Total

100

Approaches to Teaching and Learning:
Lectures, tutorials, enquiry and problem-based learning.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
Not applicable to this module.

Assessment Strategy
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Class Test: In class problem based tests Varies over the Trimester n/a Standard conversion grade scale 40% No

20

Continuous Assessment: Take home assignments Varies over the Trimester n/a Standard conversion grade scale 40% No

20

Examination: End of semester exam 2 hour End of Trimester Exam No Standard conversion grade scale 40% No

60

Carry forward of passed components
No

Resit In Terminal Exam
Spring Yes - 2 Hour
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Not yet recorded.

Name Role
Professor Adrian Ottewill Lecturer / Co-Lecturer
Assoc Professor Barry Wardell Lecturer / Co-Lecturer
Timetabling information is displayed only for guidance purposes, relates to the current Academic Year only and is subject to change.

Autumn

Lecture Offering 1 Week(s) - 1, 2, 3 Fri 10:00 - 10:50
Lecture Offering 1 Week(s) - 4, 5, 6, 7 Fri 10:00 - 10:50
Lecture Offering 1 Week(s) - 8, 9, 10, 11, 12 Fri 10:00 - 10:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Thurs 13:00 - 13:50
Lecture Offering 1 Week(s) - Autumn: All Weeks Tues 13:00 - 13:50
Autumn