## MAS336 Differential Geometry

 Semester 1, 2017/18 10 Credits Lecturer: Dr Simon Willerton Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

Differential geometry is the study of geometric objects using calculus, and it has plenty of applications in other sciences and engineering. In this introductory course, the geometric objects of our interest will be curves and surfaces. You will learn more about such familiar notions as arc lengths, angles and areas. You will also learn how to quantify the "shape" of an object, via various notions of curvature. There are rich interactions between curvature and other geometric quantities, as illustrated most notably by Gauss' Theorem and the Gauss-Bonnet Formula. For example, we can make a map of the Earth that correctly represents either all angles or all areas; but by Gauss' Theorem, the Earth's curvature prevents us from ever making a map that correctly represents distances. The Gauss-Bonnet Formula has a "local version", which computes the sum of angles in a triangle on a surface, as well as a "global version", which reveals a far-reaching connection between some small- and large-scale geometric behaviours.

Prerequisites: MAS211 (Advanced Calculus and Linear Algebra)
No other modules have this module as a prerequisite.

## Outline syllabus

Curves in R2
• basic notions and examples
• curvature
Surfaces in R3
• basic notions and examples
• metric quantities
• curvature
• parallel vector fields and geodesics
• Gauss' Theorem and the Gauss-Bonnet Formula

## Office hours

Tuesday 1.30-2.30 Hicks J19

## Aims

• to introduce differential geometry: its goals, techniques and applications;
• to translate intuitive ideas into mathematical concepts that allow quantitative studies and development of sophisticated results;
• to illustrate geometric concepts and results through many examples.

## Learning outcomes

• express geometric quantities in different ways, for conceptual or computational purposes
• curves in R2: understand the meaning and significance of curvature
• surfaces in R3: understand the meaning of curvature and its effects on other geometric quantities
• appreciate two landmark results: Gauss' Theorem and the Gauss-Bonnet Formula
• discover a number of applications of differential geometry

## Teaching methods

lectures, problem solving

20 lectures, no tutorials

## Assessment

one 2.5 hour written exam, 4 questions out of 4

## Full syllabus

Introduction (1 lecture)
Sum of angles in a triangle on a plane, a sphere, and a general surface.

I. Curves in R2 and their Curvature (4 lectures)
Parametrisation. Tangent vectors. Arc lengths. Unit-speed parametrisation. Smooth curves. Curvature. How curvature characterises the shape of a curve.
II. Surfaces in R3 (2 lectures)
Parametrisation. Tangent planes. Smooth surfaces. Maps of surfaces.
III. Metric Quantities of a Surface (3 lectures)
First fundamental form. Local isometries. Conformal maps. Area-preserving maps.
IV. Curvature of a Surface (5 lectures)
Normal vectors. Second fundamental form. Weingarten map. Normal curvature. Principal curvatures & vectors. Mean & Gaussian curvatures, Gauss' theorem
V. Parallel Vector Fields and Geodesics (3 lectures)
Vector fields along a curve. Covariant differentiation. Parallel vector fields. Geodesic curvature. Geodesics.
VI. Gauss' Theorem and the Gauss-Bonnet Formula (2 lectures)
Local Gauss-Bonnet Formula. Triangulation. Euler characteristic. Global Gauss-Bonnet Formula.