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Differential Geometry I
Last Updated: 2026-02-05 16:15:20
Abstract
Introduction to differential manifolds and differential geometry.Introduce the language, tools, and basic results of differentiable manifolds, tensors, Riemannian geometry, and related geometric structures. Relate geometric intuition to formulas involving curvature, derivatives and tensors.
Objective
Learn to compute, describe, prove, and solve problems in the language of differential geometry.
Content
Submanifolds of R^n, immersions, submersions, and embeddings, Sard's Theorem, abstract differentiable manifolds, charts, vector fields and flows, vector bundles, tensor fields, covariant derivatives, parallel transport, Riemannian metrics, geodesics, Riemann curvature tensor. Complete manifolds, Hopf-Rinow theorem. Many examples including curves, surfaces, hyperbolic space, S^3, the unit quaternions, the Gauss-Bonnet theorem, etc.
Resources
Literature
John M. Lee: Introduction to Smooth Manifolds John M. Lee: Introduction to Riemannian Manifolds This following books were inherited from before. The only one I know is DoCarmo. - Manfredo P. do Carmo: Differential Geometry of Curves and Surfaces - S. Montiel, A. Ros: Curves and Surfaces - S. Kobayashi: Differential Geometry of Curves and Surfaces - Wolfgang Kühnel: Differentialgeometrie. Kurven-Flächen-Mannigfaltigkeiten - Dennis Barden & Charles Thomas: An Introduction to Differential Manifolds
Learning Materials (Links)
- Main link
- Information
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- written 180 minutes
- Aids
- None
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Differential Geometry I |
|
4 h weekly |
| exercise |
Differential Geometry I
Groups are selected in myStudies.
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|
1 h weekly |
Offered In
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Core Courses (For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 14 of the required 26 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields.)
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Bachelor Core Courses: Pure Mathematics (Further restrictions apply, but in particular: 401-3531-00L Differential Geometry I can only be recognised for the Master Programme if 401-3532-00L Differential Geometry II has not been recognised for the Bachelor Programme. Analogously for: 401-3461-00L Functional Analysis I - 401-3462-00L Functional Analysis II 401-3001-61L Algebraic Topology I - 401-3002-12L Algebraic Topology II 401-3132-00L Commutative Algebra - 401-3146-12L Algebraic Geometry For the category assignment take contact with the Study Administration Office ( ) after having received the credits.)
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