Riemann's \textit{Commentatio Mathematica}, a reassessment (Q2928241)

Riemann's \textit{Commentatio Mathematica} is one of his more obscure papers. It was written as an entry for a prize competition of the Parisian Academy of Sciences on the subject of heat diffusion, and it deals with quadratic differential forms and their transformations. Quadratic forms were introduced as a way to deal with inhomgeneous media, the transformations were aimed at determining if the form can be reduced to one appropriate to homogeneous media. Early commentators, from Dedekind and Weber to Levi-Civita, connected it to Riemann's major paper on differential geometry. They noted that Riemann referred to that paper (at the time still unpublished) in his \textit{Commentatio}, and argued that it was reasonable to suppose that Riemann saw geometry as the conceptual underpinning of the investigation. Later commentators, \textit{R. Farwell} and \textit{C. Knee} [Hist. Math. 17, No. 3, 223--255 (1990; Zbl 0743.01017)] and \textit{K. Reich} [Arch. Hist. Exact Sci. 44, No. 1, 77--105 (1992; Zbl 0767.01023)], however, have argued that the paper should be seen as belonging to tensor analysis and that it is an anachronism to connect it to Riemann's work on geometry.NEWLINENEWLINEThe author argues strongly for the earlier interpretation, and draws attention to a well-known difficulty with the \textit{Commentatio} that concerns an equality Riemann presented without sufficient explanation. In the 19th century, Lipschitz and Beez had attempted to verify Riemann's claim, by reinterpreting it, but Levi-Civita objected that one side of the equality would seem always to be zero, invalidating Riemann's claim. The disputed point concerns second-order differentials of two kinds, and the author uses Beez's account to defend Riemann's claim more simply by supposing that Riemann was (tacitly) using normal coordinates, and then rashly generalised it to arbitrary systems. The author concludes that the original differential-geometric reading of the \textit{Commentatio} was rather close to the truth, and that its influence, through the early commentaries upon it, was certainly geometrical.