Welcome to MAT4510

The course will be based on a new set of lecture notes extending those from last year (with some changes). The first few sections will be available before the semester begins. Here is the plan for the course:

Chapter 1.  Manifolds, tangent spaces, smooth maps

Chapter 2.  Classical surface theory: First and second fundamental forms, Gauss curvature

Chapter 3.  Intrinsic geometry:  Covariant derivatives and curvature, geodesics, the exponential map

Chapter 4.  Hyperbolic Geometry:  Four models for the hyperbolic plane, geodesics, isometry groups, hyperbolic trigonometry.

Chapter 5.  The Gauss-Bonnet theorem.

The choice of material is much influenced by the book

Christian B?r: "Elementary Differential Geometry".

However, we will prove the version of the Gauss-Bonnet theorem given in Bj?rn Jahren's book, using differential forms.

-Kim Fr?yshov