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Prospectus

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Differentiable manifolds 2 (BM)

Course
2020-2021

Admission requirements

Differentiable manifolds 1 (required)
Topology (required)
Linear Algebra 1,2 (required)
Analysis 2,3 (required)
Algebra 1 (required)
Algebra 2 (recommended)
Complex Analysis (recommended)

Description

In this course, we introduce abstract smooth manifolds. This notion generalises that of a smooth surface in three dimensional space, which serves as a motivating example. We discuss the main properties of and structures on manifolds, such as vector bundles, vector fields and derivations, flows, differential forms and orientation and integrals. We then prove Stokes’ theorem and the related Gauss theorem and divergence theorem. This paves the way for a discussion of DeRham cohomology and its fundamental properties. We conclude with the basic notions of Riemannian geometry such as metrics on vector bundles and the Riemannian distance function.

Course objectives

Obtain a thorough understanding of the geometry and topology of manifolds. Develop the ability to carry out explicit calculations of differential and integral calculus on manifolds in a concrete setting. Apply these techiques in examples from geometry and dynamical systems.

Mode of instruction

weekly lectures and exercise classes

Literature

John. M. Lee, “Introduction to smooth manifolds”

Assessment Method

Written exam.

The final grade is the maximum of
-the written exam
-the weighted average of the homework (25%) and the written exam (75%)

Website

See the Brightspace page for all course information

Contact

Bram Mesland, b[dot]mesland[at]math[dot]leidenuniv[dot]nl