Homepage summer semester 2020 Differential geometric structures on manifolds with applications to General Relativity

General

The lecture will take place 2 hours per week and will contain no exercises. It is primarily meant to be an accompanying lecture for that on General Relativity, though it is really a "stand alone" and can be attended independently.

Lecture (start April 22. 2020)

Wednesday 14:15-15:45 Uhr (online)

Modules / ECTS

Bachelor programme: Wahlbereich, Moderne Aspekte der Physik.
Masters programme: Fortgeschrittene Vertiefungsphase.
ECTS points: 2

General

Methods of modern differential geometry have proven useful in almost all disciplines of physics, from analytical mechanics, thermodynamics, hydrodynamics and theory of deformable media, to theories of fundamental interactions (of Yang-Mills type) and gravity (General Relativity).

In this lecture we will first define carefully the notion of a differentiable manifold und then discuss the relevant structures on them. Special attention is given to semi-Riemannian manifolds and their use in General Relativity.

Topics

1. Topological manifiolds and differential structures.
2. Tangent- cotangent, and general tensor bundles over differentiable manifolds.
3. Exterior and Lie derivative.
4. Riemannian and semi-Riemannian structures.
5. Connections and covariant derivative.
6. Torsion and Curvature. Various derived curvature quantities (Riemann, Ricci, scalar, conformal, projective).
7. Cartan's structure equation and its use for computing the connection coefficients and Riemannian curvature tensor.
8. Affine, projective, and conformal structures; theorem of H. Weyl.
9. Fermi-Walker derivative and its physical relevance.

Literature

Manuscripts

I have a detailed 160-page manuscript (in german) the second part of which I will follow very closely. The first part is an introduction into curves and surfaces in 3-dimensional space and may serve to train your intuition. The pdf-file of that manuscript can be downlowaded here. Alternatively, if you cannot read a german text, I recommend the summary on differential geometry in the book on General Relativity by Norbert Straumann (Springer Verlag, 2013). A copy of that part of the book can be found here. For reasons of copyright I cannot provide a copy of the entire book.

Classic texts

1. Ivan Kolář, Peter W. Michor und Jan Slovák: Natural Operations in Differential Geometry (Springer Verlag, Berlin, 1993). Very concise text concerning general differential geometric structures. Fairly steep and no applications. kaum Anwendungen.
2. Shoshichi Kobayashi und Katsumi Nomizu: Foundations of Differential Geometry, Vol I and II (John Wiley and Sons)
3. Barrett O'Niel: Semi-Riemannian Geometry - with applications to Relativity Academic Press 1983
4. Detlev Laugwitz: Differentialgeometrie, B.G. Teubner (Struttgart 1977). Old fashioned but very good in its discussion of curves and surfaces in 3d space. Contains a nice proof of the Helmtoltz-Weyl theorem in the positive definite case.
5. Michael Spivak: Differential Geometry, Vol. I-V, Publish or Perish, Inc., Wilmington Delaware, 1970. 5-Volumes in which you find many things that were cut in more modern rationalised texts. Here intelligibility comes before conciseness.