# 2019/20 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 35

Module manager: Dr Gerasim Kokarev
Email: G.Kokarev@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2019/20

### Pre-requisites

 MATH2051 Geometry of Curves and Surfaces

### This module is mutually exclusive with

 MATH3113 Differential Geometry

This module is approved as an Elective

### Module summary

The course builds on the prerequisite module MATH2051 (Curves and Surfaces). The central topic is a rigorous definition and study of higher dimensional surfaces in Euclidean spaces, also known as submanifolds. In particular, we focus on understanding the basics of calculus on surfaces, study geometric quantities that are invariant under the deformations of surfaces, and learn how to distinguish between intrinsic and extrinsic properties of surfaces. We also learn basics of the covariant derivative formalism on surfaces and discuss the notion of curvature tensor. The latter theme underpins much of modern mathematics, and is ubiquitous in other disciplines, such as mathematical physics.

### Objectives

On completion of this module, students should be able to:
(a) construct Euclidean isometries
(b) calculate rotation indices of plane curves
(c) recognise regularly homotopic curves
(d) recognize when a level set defines a surface and describe its properties
(e) compute the various curvature quantities of hypersurfaces in n-space
(f) understand and recognise properties of isometries
(g) appreciate the difference between intrinsic properties and extrinsic properties
(h) prove the major results of the module, where proofs have been given

### Syllabus

To gain insights into the circle of ideas that we plan to develop, we first review and study the geometry of the Euclidean space and global theory of curves in Euclidean space. The central result in this part of the course is the Whitney-Graustein theorem; it says that two closed plane curves can be deformed into each other as soon as a single number (the so-called rotation index) is the same for both curves. We also see that this topological invariant (rotation index) is closely related to the geometry of curves. We study analogous relationships for space curves as well.

We continue with the rigorous definition and study of higher dimensional surfaces in Euclidean spaces. In particular, we learn to view surfaces as level sets of functions, and meet the Regular Value Theorem. We also develop calculus on surfaces; for example, we give precise meaning to the notion of tangent space and learn how to differentiate maps between surfaces. As important examples we study matrix groups.

In the final part of the course, we introduce the notion of isometry between surfaces, and learn what properties of a surface are intrinsic, that is, can be measured by the inhabitants of the surface without going outside it. We generalize various notions of curvature to higher dimensional surfaces, and see that for example, principal curvatures are not intrinsic. This fact is contrasted with Theorema Egregium -- it says that the Gauss curvature, defined as a product of principal curvatures, of a 2-dimensional surface is intrinsic.

We also develop a formalism of covariant derivative and introduce the curvature tensor -- the ultimate intrinsic invariant "containing all possible curvature quantities". We study its symmetries, and meet Gauss equations, establishing the relationship with the shape operator.

The covered topics include:
- Plane curves: regular homotopy, rotation index, and Whitney-Graustein theorem.
- Plane curves: isoperimetric inequality.
- Notion of a regular n-dimensional surface (submanifold) in Euclidean spaces.
- Surfaces as level sets and regular value theorem.
- Tangent spaces and differentials of maps between surfaces.
- Isometries; intrinsic and extrinsic properties of surfaces.
- Gauss map and curvature quantities.
- Theorema Egregium.

Further topics will be drawn from the following, or similar, as time allows:
- Geometry of Euclidean space; isometries and orthogonal transformations
- Space curves: congruence, Fenchel's theorem, total curvatute of a knot
- Curves on surfaces; geodesics
- Gauss Lemma and local minimising property of geodesics
- Matrix groups as regular n-dimensional surfaces; their tangent spaces.
--Vector fields and covariant derivative.
- Riemann curvature tensor, Gauss equations.
- Gauss-Bonnet Theorem.
- Abstract Riemannian metrics.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem sheets

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated