# Geometry

Lecturer Roland van der Veen, assistants: Martijn Kluitenberg and Oscar Koster. Please send all your email to Meetkunde20@gmail.com

## Course description

Traditionally geometry refers the properties of straight lines, circles, planes, angles and distances as summarized in Euclid's Elements. Loosening Euclid's axioms one finds there exist many interrelated theories of geometry, each with its own distinct interpretation of the above elementary notions of lines, distance etc. For example, what if distances play no role? (Affine geometry). What if parallel lines are to meet 'at infinty'? (projective geometry) What if we are on the sphere or even just on some potato? (differential geometry) Following Felix Klein we focus on the unifying concept of 'isometries': the motions or symmetries that preserve all the relevant geometric properties of the theory.

In this course we will introduce various theories of geometry using linear algebra as a foundation. Before studying differential geometry of curves and surfaces we will consider aspects of affine, Euclidean and possibly projective geometry.

There are two parallel problem sessions on Wednesday, the first in EA 5159.0062 for students with surnames A-M and the second for the rest. On Tuesday morning there is an additional problem session in 5161.0041b (5161.0222 on march 31, april 7) for everyone.

## Schedule

Date, location Material covered Exercises
WEEK 1: Affine space
Tue 4-2, NB 5111.0022 lecture on affine space, subspace and maps Pappus theorem lecture notes sec 2, p.1,2 (Audin ch 1)
Wed 5-2, 15-17, EA 5159.0062, NB 5114.0004 Problem session Exercises: Tutorial exercises 1 with solutions by Oscar and Martijn.
Wed 5-2, 17-19, BB 5161.0253 Lecture: Proof of Pappus theorem using dilations, affine maps lecture notes most of section 2.
WEEK 2: Euclidean space and its isometries
Tue 11-2, 9-11, 5161.0041b Problem session Exercises: Tutorial exercises 2
Tue 11-2, 15-17, NB 5111.0022 lecture on Euclidean geometry, isometries, reflections Lecture notes sec. 3.1, (Audin ch 2)
Wed 12-2, 15-17, EA 5159.0062, NB 5114.0004 Problem session, Lecture notes sec 3.1: 1-3. Exercises: Tutorial exercises 3
Wed 12-2, 17-19, BB 5161.0253 Lecture: Lecture notes sec. 3.2. Homework 1, with solutions (due Wednesday 26-2 before the problem session): Lecture notes sec. 2:10,11 and sec 3:5,6,7
WEEK 3: Plane Euclidean geometry
Tue 18-2, 9-11, 5161.0041b Problem session Exercises: Tutorial exercises 4
Tue 18-2, 15-17, NB 5111.0022 Lecture: Lecture notes sec. 4.1 (Audin Ch 3) Plane Euclidean geometry, isometries, angles.
Wed 19-2, 15-17, EA 5159.0062, NB 5114.0004 Problem session Exercises: Tutorial exercises 5.
Wed 19-2, 17-19, BB 5161.0253 Lecture: Angles in the plane and their measures. Lecture notes sec. 4.2 Orientation and convexity.
WEEK 4: Spatial Euclidean geometry
Tue 26-2, 9-11, 5161.0041b Problem session Exercises: Tutorial exercises 6
Tue 26-2, 15-17, NB 5111.0022 Lecture: 3d Euclidean geometry, rotations, orientation, convexity, Platonic solids. Audin Chapter 4
Wed 27-2, 15-17, EA 5159.0062, NB 5114.0004 Problem session Exercises: Tutorial exercises 7
Wed 27-2, 17-19, BB 5161.0253 Lecture: Spherical geometry, Euler formula, dodecahedron.. Homework 2 (due Wednesday 11-3 before the problem session)
WEEK 5: Riemannian geometry
Tue 3-3, 9-11, 5161.0041b Problem session Exercises: Tutorial exercises 8
Tue 3-3, 15-17, NB 5111.0022 Lecture: Riemannian metrics and charts, pull-back metric. Derivative, chain rule, partial derivative.
Wed 4-3, 15-17, EA 5159.0062, NB 5114.0004 Problem session Exercises: Tutorial exercises 9
Wed 4-3, 17-19, BB 5161.0253 Lecture: Length of a curve, angles in Riemannian charts Examples of Riemannian charts, pull-back metrics, isometries.
WEEK 6: Variational calculus and geodesics
Tue 10-3, 9-11, 5161.0041b Problem session Exercises: Tutorial exercises 10
Tue 10-3, 15-17, NB 5111.0022 Lecture: Variational calculus
Wed 11-3, 10-12, LB 5173.0149 ADDITIONAL CATCH UP SESSION
Wed 11-3, 15-17, EA 5159.0062, NB 5114.0004 Problem session Exercises: Tutorial exercises 11
Wed 11-3, 17-19, BB 5161.0253 Lecture: Geodesic equation, ordinary differential equations Homework 3 (due Monday 30-3 before the problem session)
WEEK 7: Geodesics, covariant derivative
Exercises: Tutorial exercises 12
Tue 17-3, 15-17, Nestor, BB collaborate LIVE ONLINE Lecture: Geodesics on surfaces in Euclidean space.
Wed 18-3, 15-17, Nestor, BB collaborate LIVE ONLINE Lecture: Covariant derivative
WEEK 8: Riemannian curvature
Exercises: Tutorial exercises 13
Tue 24-3, 15-17, Nestor LIVE ONLINE Lecture: Differentiating vector fields and Levi-Civita connection
Wed 25-3, 15-17, Nestor LIVE ONLINE Lecture: Riemannian curvature
TAKE HOME EXAM Will be posted on Tue 7-4 at noon, and should be handed in Latex format on Tue 21-4 before noon. Solutions are now available
REEXAMINATION The resit will consist of a 30 minute online oral exam. You are expected to know the basic definitions and theorems and we can also ask you questions about the homework assignments and the questions from the take-home exam so make sure you have reviewed those before the oral exam. If you would like to participate in the resit, please send us an email at meetkunde20@gmail.com before June 15. We will then proceed to schedule the oral exams in the week of June 30 (first week of July).

## Assessment

The final grade is computed as max((0.3H + 0.7T), T), where H is the score for homework and T is the score for the take home exam. The homework consists of three problem sets and your online activity on the nestor forum for the course can inprove the homework grade. More precisely you can add up to one point to your average homework grade by participating in the Nestor discussion forum until March 14. After March 14 you can again add one more point to your average homework by participating in the forum until the exam date. The final exam will contain at least one exercise that was discussed in the problem sessions.

## Prerequisites

Linear algebra 1,2 and Calculus 1,2 are essential.

## Literature

Lecture notes based on parts of the book 'Geometry' by Michele Audin will be used and 'Differential geometry of curves and surfaces' by Manfredo do Carmo is recommended for background material. For background in linear algebra the book 'Linear algebra' by K. Janich is recommended. Tex file also available.

## Geogebra illustrations

Constructions in plane geometry can be visualised easily using Geogebra.

## Mathematica illustrations

Differential geometry can be visualized and computed using Mathematica.