Cartan, Schouten and the search for connection. (English) Zbl 1397.01020

The authors give a detailed mathematical exposition of a 1925–1926 collaboration between Élie Cartan and Jan Arnoldus Schouten based on published articles and on archival correspondence in the Archives of the Académie des Sciences in Paris and Schouten’s papers at the Amsterdam Mathematical Centrum. Excerpts from this correspondence, presented in English in the text and in the French original in an appendix, support the authors’ characterization of Cartan’s and Schouten’s respective contributions to the results and publications that resulted from their collaboration. The analysis emphasizes significant differences in concepts and formalisms between Cartan and Schouten, with the former emphasizing geometrical interpretations in terms of moving frames while the latter offered a systematic, symbol-driven differential calculus of geometric groups. Together, they introduced a series of connections on manifolds and offered a classification of Riemannian manifolds admitting an absolute parallelism. Cogliati and Mastrolia identify a variety of motivations, tentative ideas, and previous results (for instance from Levi-Civita, Clifford, and Eisenhart) that shaped Cartan’s and Schouten’s research and close their discussion with an extensive technical presentation of the results in terms of more recent differential geometry. Their article is of greatest interest to those well-versed in differential geometry who wish to understand the finer technical points of a significant historical episode in the theory of connections on Lie groups, elucidated through unpublished manuscript evidence.


01A60 History of mathematics in the 20th century
53-03 History of differential geometry
53B20 Local Riemannian geometry
53B05 Linear and affine connections

Biographic References:

Cartan, Élie; Schouten, Jan Arnoldus
Full Text: DOI Link


[1] Agricola, I., The Srní lectures on non-integrable geometries with torsion, Arch. Math. (Brno), 42, Suppl., 5-84 (2006) · Zbl 1164.53300
[2] Agricola, I.; Ferreira, A. C.; Friedrich, T., The classification of naturally reductive homogeneous spaces in dimensions \(n \leq 6\), Differ. Geom. Appl., 39, 59-92 (2015) · Zbl 1435.53040
[3] Agricola, I.; Friedrich, T., A note on flat metric connections with antisymmetric torsion, Differ. Geom. Appl., 28, 4, 480-487 (2010) · Zbl 1201.53052
[4] Akivis, M.; Rosenfeld, B., Élie Cartan (1869-1951) (2011), American Mathematical Society
[5] Alías, L. J.; Mastrolia, P.; Rigoli, M., Maximum Principles and Geometric Applications, Springer Monographs in Mathematics (2016), Springer-Verlag · Zbl 1346.58001
[6] Cartan, E., La structure des groupes de transformations continus et la théorie du trièdre mobile, Bull. Sci. Math., 34, 250-284 (1910) · JFM 41.0182.01
[7] Cartan, E., Sur les équations de la gravitation d’Einstein, J. Math. Pures Appl., 1, 141-204 (1922) · JFM 48.0993.02
[8] Cartan, E., Sur les équations de structure des espaces généralisés et l’expression analytiques du tenseur d’Einstein, C. R. Acad. Sci., 174, 1104-1107 (1922) · JFM 48.0854.05
[9] Cartan, E., Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Acad. Sci., 174, 593-595 (1922) · JFM 48.0854.02
[10] Cartan, E., Sur les variétés à connexion affine et la théorie de la relativité généralisée (prémière partie), Ann. Sci. Éc. Norm. Supér., 40, 325-412 (1923) · JFM 49.0542.02
[11] Cartan, E., Les récentes généralisations de la notion d’espace, Bull. Sci. Math., 48, 294-320 (1924) · JFM 50.0589.01
[12] Cartan, E., Sur les variétés à connexion affine et la théorie de la relativité généralisée (prémière partie) (suite), Ann. Sci. Éc. Norm. Supér., 41, 1-25 (1924) · JFM 51.0581.01
[13] Cartan, E., Sur les variétés à connexion affine, et la théorie de la relativité généralisée (deuxième partie), Ann. Sci. Éc. Norm. Supér., 3, 42, 17-88 (1925) · JFM 51.0582.01
[14] Cartan, E., Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. Fr., 54, 214-264 (1926) · JFM 52.0425.01
[15] Cartan, E., La géométrie des groupes de transformations, J. Math. Pures Appl., 6, 1-119 (1927) · JFM 53.0388.01
[16] Cartan, E., Leçons sur la géométrie des espaces de Riemann (1946), Gauthier-Villars · Zbl 0060.38101
[17] Cartan, E.; Einstein, A.; Debever, R., Élie Cartan-Albert Einstein: Letters on Absolute Parallelism, 1929-1932 (1979), Princeton University Press
[18] Cartan, E.; Schouten, J. A., On the group-manifold of simple and semi-simple groups, Proc. Roy. Acad. Amsterdam, 29, 803-815 (1926) · JFM 52.0422.04
[19] Cartan, E.; Schouten, J. A., On riemannian geometries admitting an absolute parallelism, Proc. Roy. Acad. Amsterdam, 29, 933-946 (1926) · JFM 52.0744.02
[20] Chorlay, R., Géométrie et topologie différentielles (1918-1932) (2015), Hermann · Zbl 1284.53002
[21] Cogliati, A., Variations on a theme: Clifford’s parallelism in elliptic space, Arch. Hist. Exact Sci., 69, 4, 363-390 (2015) · Zbl 1331.51017
[22] Cogliati, A., Schouten, Levi-Civita and the notion of parallelism in riemannian geometry, Hist. Math., 43, 427-443 (2016) · Zbl 1354.01013
[23] Cosserat, E.; Cosserat, F., Sur la théorie de l’élasticité. Première memoire, Ann. Fac. Sci. Toulouse, 10, 3-4 (1896) · JFM 27.0684.06
[24] Cosserat, E.; Cosserat, F., Théorie des corps déformables (1909), Hermann · JFM 40.0862.02
[25] Eisenhart, L. P., Linear connections of a space which are determined by simply transitive continuous groups, Proc. Natl. Acad. Sci. USA, 11, 5, 246-250 (1925) · JFM 51.0329.03
[26] Eisenhart, L. P., Riemannian Geometry (1966), Princeton University Press · Zbl 0041.29403
[27] Kobayashi, S.; Nomizu, K., Foundations of Differential Geometry, vol. 1 (1963), New York · Zbl 0119.37502
[28] Lagrange, R., Sur le Calcul Différentiel Absolu (1923), Ph.D. dissertation · JFM 50.0673.01
[29] Lazar, M.; Hehl, F. W., Cartan’s spiral staircase in physics and, particular, in the gauge theory of dislocations, Found. Phys., 40, 1298-1325 (2010) · Zbl 1216.83043
[30] Levi-Civita, T., Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana, Rend. Circ. Mat. Palermo, 42, 1, 173-204 (1917) · JFM 46.1125.02
[31] Nabonnand, P., La notion d’holonomie chez Élie Cartan, Rev. Hist. Sci., 62, 221-245 (2009) · Zbl 1213.01039
[32] Nijenhuis, A., J.A. Schouten: a master at tensors, Nieuw Arch. Wiskd., 20, 3, 1-8 (1972)
[33] Postnikov, M., Geometry VI: Riemannian Geometry, Encyclopaedia of Mathematical Sciences (2001), Springer: Springer Berlin, Heidelberg · Zbl 0993.53001
[34] Reich, K., Die Entwicklung des Tensorskalküls: vom absoluten Differentialkalkül zur Relativitätstheorie (1994), Birkhäuser Verlag
[35] Salamon, S., Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics Series, vol. 201 (1989), Longman Scientific & Technical: Longman Scientific & Technical Harlow, Copublished in the United States with John Wiley & Sons, Inc., New York · Zbl 0685.53001
[36] (Scholz, E., Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work. Hermann Weyl’s Raum-Zeit-Materie and a General Introduction to His Scientific Work, DMV Seminars, Basel (2001))
[37] Schouten, J. A., Über die verschiedenen Arten der Übertragungen in einer \(n\)-dimensionalen Mannigfaltigkeit, die einer Differentialgeometrie zugrunde gelegt werden können, Math. Z., 13, 56-81 (1922) · JFM 48.0858.01
[38] Schouten, J. A., On a non-symmetrical affine field theory, Proc. Roy. Acad. Amsterdam, 26, 850-857 (1923)
[39] Schouten, J. A., Der Ricci-Kalkül (1924), Verlag von Julius Springer · Zbl 0403.53007
[40] Schouten, J. A., Über die Projektivkrümmung und Konformkrümmung halbsymmetrischer Übertragungen, In Memoriam Lobachevsky, 2, 90-98 (1927) · JFM 53.0687.01
[41] Struik, D. J., Theory of Linear Connections (1934), Springer: Springer Berlin · Zbl 0008.08402
[42] Struik, D. J., Schouten, Levi-Civita and the emergence of tensor calculus, (Rowe, D. M.J., The History of Modern Mathematics. Volume II: Institutions and Applications (1989), Academic Press), 99-105
[43] Tricerri, F.; Vanhecke, L., Homogeneous Structures on Riemannian Manifolds, London Mathematical Society Lecture Note Series, vol. 83 (1983), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0509.53043
[44] Wolf, J. A., On the geometry and classification of absolute parallelisms. I, J. Differ. Geom., 6, 3, 317-342 (1972) · Zbl 0251.53014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.