# Elements of Geometry： Geometry of Curves and Surfaces

Lecturer Shin NAYATANI, Professor Graduate School of Mathematics, 2012 Spring Department of Mathematics 3rd year students (6・3 hours / session One session / week 15 weeks / semester)

### 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.

### 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

#### Last updated

December 08, 2015

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