Introduction to Tensor Formalism and Differential Geometry for Physics
Instructors: Univ.-Prof. Dr. Nikolaos PapadopoulosShortname: 08.128.7034
Course No.: 08.128.7034
Course Type: Vorlesung
Requirements / organisational issues
Basic knowledge of linear algebra and calculus is assumed.The lecture is suitable for both undergraduate (Bachelor’s) and graduate (Master’s) students.
Start of the lecture: 10 Nov. 2025
Recommended reading list
Selection:- Agricola, I., Friedrich, Th., Globale Analysis, Differentialformen in Analysis, Geometrie und Physik, Vieweg, 2001 (German Edition)
- Agricola, I, Friedrich, Th.: Global Analysis: Differential Forms in Analysis, Geometry and Physics, Graduate Studies in Mathematics, Vol. 52, 2002 (English Edition)
- Papadopoulos, N.A., Scheck, F.: Linear Algebra for Physics, SpringerNature, 2024;
- Renteln, P.: Manifolds, Tensors, and Forms, An Introduction for Mathematicians and Physicists, Cambridge University Press, 2014;
Contents
The aim of this lecture is to introduce the phenomenon of curvature—as it appears in General Relativity (GR) and in gauge theories—in a direct and accessible way.
The “royal road” to this goal is through an appropriate approach to tensor formalism. This will be developed on the basis of a specially designed elementary linear algebra, formulated from the very beginning in tensor notation.
- Brief introduction to the basic elements of linear algebra, with the aim of making the treatment of tensor formalism significantly easier and more concise.
- Tensor formalism with special attention to alternating multilinear forms and the exterior product (wedge product).
- Tangent and cotangent spaces of a manifold.
- The role of vector fields in differential geometry.
- Covariant derivative and connection.
- Toward Gaussian curvature.
- Curvature in gauge theories: structure equations, Bianchi identities, and gauge transformations.
Additional information
Prominent mathematicians such as Raoul Bott, who also made important contributions to physics, emphasize that 80% of mathematics is linear algebra. This lecture makes deliberate use of that perspective to present differential geometry in a direct and focused way.Dates
| Date (Day of the week) | Time | Location |
|---|---|---|
| 10/27/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 11/03/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 11/10/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 11/17/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 11/24/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 12/01/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 12/08/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 12/15/2025 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 01/05/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 01/12/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 01/19/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 01/26/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 02/02/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |
| 02/09/2026 (Monday) | 16:15 - 17:45 | 01 128 Galilei-Raum 2413 - Neubau Physik/Mathematik |