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Prospectus

# Differentiable manifolds 1

Course
2021-2022

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

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

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