401-3531-00L Differential Geometry I
| Semester | Autumn Semester 2024 |
| Lecturers | U. Lang |
| Periodicity | yearly recurring course |
| Language of instruction | English |
| Comment | At most one of the three course units (Bachelor Core Courses) 401-3461-00L Functional Analysis I 401-3531-00L Differential Geometry I 401-3601-00L Probability Theory can be recognised for the Master's degree in Mathematics or Applied Mathematics. In this case, you cannot change the category assignment by yourself in myStudies but must take contact with the Study Administration Office (www.math.ethz.ch/studiensekretariat) after having received the credits. |
| Abstract | Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in R^n, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||||||||||||||||||
| Learning objective | Learn the basic concepts and results in differential geometry and differential topology. Learn to describe, compute, and solve problems in the language of differential geometry. | |||||||||||||||||||||
| Content | Curves, (hyper-)surfaces in R^n, first and second fundamental forms, geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet, minimal surfaces. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem. | |||||||||||||||||||||
| Lecture notes | Partial lecture notes are available from https://people.math.ethz.ch/~lang/ | |||||||||||||||||||||
| Literature | Differential geometry in R^n: - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - S. Montiel, A. Ros: Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Christian Bär: Elementare Differentialgeometrie Differential topology: - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds - Victor Guillemin & Alan Pollack: Differential Topology - Morris W. Hirsch: Differential Topology - John M. Lee: Introduction to Smooth Manifolds | |||||||||||||||||||||
Competencies |
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