Course Material

Course Structure (2015)

(35 lectures : 8+11+11+5)

 

Introduction

Part I : Curves (8)

Part II : Surfaces (11)

Part III : Curvature (11)

Part IV : Geodesics (5)

 

Lectures

Preliminaries: linear algebra and topology 

(Curves)

Lecture 1 : What is a curve ? Arc-length

Lecture 2 : Reparametrization

Lecture 3 : Level curves vs. parametrized curves

Lecture 4 : Curvature I

Lecture 5 : Curvature II

Lecture 6 : Plane curves

Lecture 7 : Space curves I

Lecture 8 : Space curves II

(Surfaces)

Lecture   9 :  What is a surface ?

Lecture 10 :  Smooth surfaces. Smooth maps

Lecture 11 : Tangents and derivatives I

Lecture 12 : Tangents and derivatives II

Lecture 13 : Level surfaces. Quadric surfaces

Lecture 14 : Ruled surfaces and surfaces of revolution

Lecture 15 : Normals (and orientability). Compact surfaces

Lecture 16 : Length of curves on surfaces

Lecture 17 : Isometries of surfaces I

Lecture 18 : Isometries of surfaces II

Lecture 19 : Conformal mappings on surfaces

(Curvature)

Lecture 20 : The second fundamental form

Lecture 21: The Gauss and Weingarten maps

Lecture 22 : Normal and geodesic curvatures

Lecture 23 : Parallel transport and covariant derivative I

Lecture 24 : Parallel transport and covariant derivative II

Lecture 25 : Gaussian and mean curvatures

Lecture 26 : Principal curvatures of a surface I

Lecture 27 : Principal curvatures of a surface II

Lecture 28 : Flat surfaces

Lecture 29 : Surfaces of constant mean curvature

Lecture 30 : Gaussian curvature of compact surfaces

(Geodesics)

Lecture 31 : Definition and basic properties

Lecture 32 : Geodesic equations I

Lecture 33 : Geodesics equations II

Lecture 34 : Geodesics on surfaces of revolution

Lecture 35 : Geodesic as shortest paths

 

Key Words and Results

Key words

Key results

Last Modified: Wed, 22 Jul 2015 11:13:23 SAST