Elements of Geometry: Geometry of Curves and Surfaces

Shin NAYATANI Professor

Department: Graduate School of Mathematics

Class Time: 2012 Spring Monday
Recommended for: Department of Mathematics 3rd year students

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Course Overview

Geometry is a field in mathematics that aims to understand the properties of shapes and spaces. In this course, students will learn some methods to understand curves and surfaces in R³ using linear algebra and calculus as an introduction to geometry.

The primary goal of this course is to understand the idea of curvature and to be able to calculate it for some examples. Different from curves; curvature is not unique for surfaces. Therefore, the next step is to understand that each curvature expresses how surfaces are curved from respective viewpoint, and especially understanding the meaning of the Theorema Egregium derived by Gauss.

The final goal of this lecture is to understand and prove the Gauss-Bonnet theorem, which relates the curvature of surfaces to the Euler characteristic. Students will also learn the differential form and Stokes’ theorem while proving the Gauss-Bonnet theorem.

Along with attempting to convey the interest of geometry as a pure science, I will also try to introduce typical curves in our daily lives and the utility of geometry. In addition, I will introduce the idea of manifold to link with the course of study that students will undertake in their 4th year.

Key Feature

The aim of this lecture is to understand curvature, which we use to quantitatively express how curves and surfaces are curved. First of all, calculating the curvature of a given curve or surface is the basis of this subject, so we took time for students to do calculations and to derive some important formulas. In addition to this, because we want students to understand the curvature conceptually, we tried to explain the process of formulating the curvature as carefully as possible.

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Course Schedule

Session Contents
1 Curves in R³ (1): definition, length and arc length parameter
2 Curves in R³ (2): curvature, torsion, and Frenet-Serret formulas
3 Curves in R³ (3): calculate curvature and torsion
4 Curves in R³ (4): meanings of curvature and torsion, fundamental theorem of curves
5 Surfaces in R³ (1): definition, tangential plane, unit normal vector, quick test
6 Surfaces in R³ (2): first and second fundamental forms, Gaussian curvature, mean curvature
7 Surfaces in R³ (3): meanings of Gaussian curvature and mean curvature
8 Surfaces in R³ (4): Theorema Egregium
9 Review & Mid-term exam
10 Surfaces in R³ (5): global surfaces
11 Geodesic curvature of curves, Gauss-Bonnet theorem
12 Differential forms and Stokes' theorem
13 Connection forms and structural equations
14 Proof of Gauss-Bonnet theorem
15 Review & Final exam

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Page last updated April 8, 2015

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.

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