4 4 6 D O C U M E N T 5 2 0 M A Y 1 9 2 9 at Mr. Hadamard’s[6] place, by taking a sphere and considering two vectors as par- allels, each making the same angle with the meridian that passes through its origin: the corresponding geodesics are the loxodromies (this example is also quoted in an article: “Sur les récents généralisations de la notion d’espace” [On Some Recent Generalizations of the Notion of Space], (Bull. Sciences Math. vol. 48, 1924, pp. 294–320).[7] With the terminology that I introduced, the spaces that have a Euclidean connec- tion include curvature and torsion in those spaces where the parallelism is defined as by Levi-Civita, the torsion is zero in those spaces where the parallelism is ab- solute (distant parallelism), the curvature is zero. These are thus spaces without curvature but with torsion. I wrote a long article that appeared in the Annales de l’École Normale (especially vol. 42, 1925) under the title: “On the Types of Affine Connections and the General Theory of Relativity” it was a systematic study of the tensors that are at the origins of either curvature or of torsion. One of them, which produces the torsion, has precisely all the mathematical characteristics of the elec- tromagnetic potential.[8] The types of Riemannian spaces with distant parallelism play an important role in group theory. I studied all that in another long article, “The Geometry of Trans- formation Groups” (J. de Math. Pures et Appliquées, vol. 6, 1927, pp. 1–119).[9] If the space is one that is representative of the transformations of a continuous group, there are in fact two remarkable affine connections without curvature (distant par- allelism) the condition by which the tensor introduced through distant parallelism has constant coefficients is precisely that the space considered should be the space of the representation of a group (simple or semi-simple if the space is Riemannian). I don’t want to bore you any more with an enumeration of the papers where the notion of torsion is utilized. I shall simply take the liberty of sending you the texts of two lectures, one of them given in Toronto in 1924,[10] the other in Bern in 1927,[11] where I explained my general theory without using any formulas. It goes well beyond the domain of Riemannian geometry in particular, I permit myself to call your attention to what was said in the first lecture on pages 92–93 about the geometrical space of your first theory of relativity and also in the second, on p. 209, where I spoke explicitly about absolute parallelism, and p. 217, where I intro- duced the two absolute parallelisms of the space of a group. Please excuse, dear sir, this too long letter, with my sincere regards and admira- tion, E. Cartan The article “On the Types of Affine Connections and the General Theory of Rela- tivity” appeared in the Annales de l’École Normale, vol. 40, 1923, pp. 325–412 vol. 41, 1924, pp. 1–29 vol. 42, 1929, pp. 17–88.[12]