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

Cogliati, A.; Mastrolia, P.

Hist. Math. 45, No. 1, 39-74 (2018).

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technical points of a significant historical episode in the theory of connections on Lie groups, elucidated through unpublished manuscript evidence.

Reviewer: Michael Barany (Hanover)

MSC:

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

Keywords:

Lie groups; connection theory; Riemannian geometry; absolute parallelism; epistolary mathematics

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