Welcome to the wonderful world of Geometry!

This course is an introduction to differential and Riemannian geometry: a beautiful language in which much of modern mathematics and physics is spoken. We will address questions like what is the possible shape of the universe? What is a "straight line" in a general space? and what can it tell us about that space? We will cover

• The geometry of Curves in Euclidean space,
• Manifolds in Euclidean space,
• The geometry of surfaces in Euclidean space,
• Intrinsic geometry of surfaces,
• Influences of curvature on topology.

• Minimal surfaces and surfaces of constant mean curvature (Soap bubbles),
• Analysis on surfaces,
• Higher dimensional manifolds.

Lecture Notes:

1. Introduction
1. What is differential Geometry
2. The geometry of Euclidean spac (Tapp Chapter 1.8)

2. Curves
1. Curves in R^n (Tapp Chapter 1.3 to 1.5, do Carmo Chapter 1-2, 1-3, and 1-5)
2. Curves in R^2
1. Local theorey: signed curvature (Tapp Chapter 1.6)
2. Rotation number, total curvature, and regular homotopy (Tapp Chapter 1.6, do Carmo Chapter 1-7)
3. Convexity and curvature (Tapp Chapter 2.2 and 2.3, do Carmo Chapter 1-7)
4. Length, width, and area (Tapp 2.5, do Carmo 1-7, do Carmo Chapter 1-7)
3. Curves in R^3 (Tapp Chapter 1.7, do Carmo Chapter 1-5 and 1-6)

3. Manifolds
1. Recollections from Calculus (Tapp Chapter 3.1)
2. Manifolds in R^n (partially Tapp Chapter 3.2 and 3.3, do Carmo Chapter 2.2)

4. Surfaces
1. Geometry of surfaces in R^3 (Tapp Chapter 4.1 and 4.2, do Carmo Chapter 3-2 and 3-3)
2. Local coordinates, curvature, and area (Tapp Chapter 3.6, 3.7, and 4.5, do Carmo Chapters 2-5, 3-3)
3. Some implications of curvature

5. Intrinsic Geoemtry of Surfaces
1. What is Intrinsic Geometry? (Tapp Chapter 5, do Carmo Chapters 4-1 and 4-3)
2. Covariant derivatives and parallel transport (Tapp Chapter 5, do Carmo Chapter 4-4)
3. Geodesics (Tapp Chapter 5, do Carmo 4-4)

6. Curvature and Surfaces
1. Classification of Surfaces (Tapp Chapter 6)
2. Gauss-Bonnet Theorem (Tapp Chapter 6.1 and 6.2, do Carmo Chapter 4-5)
3. Geodesic coordinates
4. Prescribing curvature
7. Mean Curvature

Course Information:

• Information about the class.
• Homework Assignments.
• Contact me at etnyre "at" math.gatech.edu.