On the genesis of the concept of covariant differentiation.

*(English)*Zbl 0876.01030The covariant derivative was explicitly introduced by G. Ricci-Curbastro in 1887. It arose from two traditions on ‘invariance’ running through the 19th century, one algebraic or algorithmic, the other analytical, and partly variational. The role of the algebraic theory of invariants had originated with Christoffel who was influenced by the publication (in 1868) of Riemann’s lecture on geometry, more precisely by the problem of equivalence of quadratic differential forms. Much of this had already been touched upon in a geometrical paper of Casorati (1860) who developed an elimination theory and aimed at a systematic method to find differential invariants (funzione inalterabile) in surface theory. Lipschitz used a mixed (algebraic cum variational) approach. The analytical tradition is traced back to Lamé (1834/59) and his differential parameters, taken up by Beltrami (1864), and aiming at differential operators in curvilinear coordinates. The author claims that this was the main incentive for Ricci to reconsider expressions already present in the papers of Christoffel, and to interpret them as generalized derivatives. Ricci used elimination theory merely as a convenient method. Indeed, the formal definition came only after prior employment of the expressions in the theory of partial differential equations. In the work of Ricci the notion of a covariant derivative conceptually anticipated and implied that of the tensor field that began to appear only in 1888.

[Reviewer’s comment: In contrast to all this, K. Reich finds the main roots of Ricci’s calculus in the algebraic theory of invariants [see p. 213 of K. Reich, Die Entwicklung des Tensorkalküls. Vom absoluten Differentialkalkül zur Relativitätstheorie. Basel: Birkhäuser (1994; Zbl 0820.01009)]. Taken together, the researches of Reich and the author refute convincingly the often claimed influence of differential geometry on the invention of tensor analysis].

[Reviewer’s comment: In contrast to all this, K. Reich finds the main roots of Ricci’s calculus in the algebraic theory of invariants [see p. 213 of K. Reich, Die Entwicklung des Tensorkalküls. Vom absoluten Differentialkalkül zur Relativitätstheorie. Basel: Birkhäuser (1994; Zbl 0820.01009)]. Taken together, the researches of Reich and the author refute convincingly the often claimed influence of differential geometry on the invention of tensor analysis].

Reviewer: Detlef Laugwitz (Darmstadt)