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Differentiable manifolds 1


Admission requirements

Linear Algebra 1,2,
Analysis 2,3,
Complex Analysis,
Algebra 1.


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

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%).


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