Abstract
This chapter introduces the theory of connections and metrics. It includes an extensive historical account of the development of mathematical and physical ideas which eventually lead to both general relativity and modern gauge theories of the non-gravitational interactions. The notion of geodesics is introduced and exemplified with the detailed presentation of a pair of examples in two dimensions, one with Lorentzian signature, the other with Euclidian signature.
My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful…
Hermann Weyl
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Notes
- 1.
The translation of Riemann’s essay from German into English was done by William Clifford.
- 2.
In the original German text of Riemann these were named mehrfach ausgedehnter Grossen. In modern scientific German the notion of manifolds is referred to as mannigfaltigkeiten.
- 3.
Einfachsten Thatsachen in the original German text.
- 4.
He gave to science Absolute Differential Calculus, essential instrument of the Theory of General Relativity, a new vision of the Universe.
- 5.
The very first embryonal idea of the Ricci tensor actually appeared as early as 1892 in another publication of its inventor [7].
- 6.
Ludwig Maurer (1859–1927) obtained his Doctorate in 1887 from the University of Strasbourg (at the time under German rule after the defeat of France in the 1870 war) and became professor of Mathematics at the University of Tübingen. His doctoral dissertation Zur Theorie der linearen Substitutionen [20] happens to contain a germ of the idea of Maurer-Cartan forms developed by Cartan in 1904.
- 7.
About the functions which help determining the attraction of general spheroids. Programme for a thesis about some properties of curves with a double curvature.
- 8.
We shall discuss this notion at length in Volume 2 in the chapters dealing with symmetries of black-holes and cosmological solutions.
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Frè, P.G. (2013). Connections and Metrics. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5361-7_3
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