RIEMANNIAN GEOMETRY - 2020/1

Module code: MAT3044

Module Overview

The module introduces the subject of manifolds of dimension greater than two, with particular emphasis on manifolds which admit a metric which enables the measurement of distance. The course begins with the introduction of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces, and Lie groups. The course continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties.

Module provider

Mathematics

GUTOWSKI Jan (Maths)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

JACs code: G100

Module cap (Maximum number of students): N/A

Module Availability

Semester 2

Prerequisites / Co-requisites

MAT2047

Module content

Indicative content includes:

• Manifolds and Vector Fields: (2.5 weeks) Abstract manifolds, with examples such as spheres and Lie groups. Tangent vectors and vector fields, integral curves. The vector field commutator and its geometric interpretation. Jacobi identity. Lie derivatives.

• Metrics, geodesics and isometries: (2.5 weeks) Metrics and pull-back Metrics, with examples. Isothermal co-ordinates. Derivation of geodesic equations. Examples of isometries and application to solving the geodesic equations.

• Covariant Derivative: (3 weeks) Aﬃne connections, torsion, Christoﬀel symbols, the Levi-Civita connection. Parallel transport with application to geodesics.

• Curvature: (3 weeks) Curvature tensor, algebraic and geometric properties of Riemann curvature. Ricci tensor and scalar. Low dimensional examples, relation to Gauss curvature of surfaces in 3 dimensions. Curvature of hypersurfaces and Einstein manifolds.

Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20
Examination EXAMINATION 80

Alternative Assessment

N/A

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

• The ability to understand and formulate statements about the geometry of various surfaces in higher dimensional space.

• A knowledge of the subject, important definitions, and concepts.

• The ability to apply this knowledge to the analysis of unseen problems in the tests and the examination.

Thus, the summative assessment for this module consists of:

• One two hour examination (the best three answers out of four questions contribute to the exam mark) at the end of Semester 2; worth 80% of the module mark.

• One in-semester test, worth 20% of the module mark.

Formative assessment and feedback

There is feedback from marked coursework assignments over an 11 week period. Verbal feedback is provided by the lecturer during seminars, and also in office hours.

Module aims

• Introduce students to the geometry of manifolds of dimension greater than two, with the added feature of a metric on the manifold.
• Introduce the students to the concepts of a covariant derivative and connection and their implications.
• Analysis of the above new concepts in the setting of examples like hypersurfaces, homogeneous spaces, and projective spaces.

Learning outcomes

Attributes Developed
1 On successful completion of this module, students should understand the fundamental properties of Riemannian geometry, including the concept of a manifold, the use of a metric, the definition of a covariant derivative and its relation to a connection, the fundamental theory of Riemannian geometry, isometries, geodesics, parallel transport, and the various curvatures of a Riemannian manifold.  KC
2 On successful completion of the module the students should have a clear idea of how to implement the above theory in examples.  KCP

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

Independent Study Hours: 117

Lecture Hours: 33

Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Be an introduction to the core ideas in Riemannian geometry, concentrating in particular on metric, covariant derivative, and curvature and how they can be used to identify properties and types of manifolds.

The learning and teaching methods include:

• 2x1 and 1x1 hour lectures per week, for 11 weeks, with typeset notes to complement the course, and Q+A opportunities for students.

• 4 dedicated lectures for example classes to discuss aspects of the examples and their solution.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.