## Admission requirements

Linear Algebra 1,2,

Analysis 2,3,

Complex Analysis,

Algebra 1.

## Description

In this course, we will focus on curves and surfaces embedded into three-dimensional space. We begin the course with the basics of curves: curvature and torsion. Then, the basics of surfaces in three-dimensional space will be discussed: the First and Second Fundamental Form, the Gauss map, and the principal curvatures. We continue the course with the intrinsic geometry of surfaces proving Gauss-Bonnet Theorem. We end the course with the definition of topological manifolds and with the classification of orientable compact topological surfaces.

## Course objectives

- Multivariate differentiation: implicit function theorem, inverse function theorem
- Curves: Parametrized curves, curvature, canonical form and global properties of plane curves
- Surfaces: Regular surfaces, tangent planes, fundamental forms, orientation, Gauss-map, parallel-transport, geodesics, Gauss-Bonnet theorem
- Introduction to Topological Manifolds: Classification of surfaces, Whitney's embedding theorem

## Mode of instruction

weekly lectures

## Assessment method

Weekly lectures and problem sessions. Written exam plus weekly homework. The final grade is the weighted average of the written exam (80%) and the homework grade (20%).

## Literature

Manfredo P. Do Carmo: Differential Geometry of Curves and Surfaces

## Brightspace/website

https://sites.google.com/site/mhablicsek/teaching

## Contact

mhablicsek[at]gmail.com