5 Differential Geometry’s High Tide

In the introduction to his book, Struik distinguished three directions in the development of the theory of linear connections [337]:

  1. The generalisation of parallel transport in the sense of Levi-Civita and Weyl. Schouten is the leading figure in this approach [300Jump To The Next Citation Point].
  2. The “geometry of paths” considering the lines of constant direction for a connection – with the proponents Veblen, Eisenhart [122Jump To The Next Citation Point114115373], J. M. Thomas [348], and T. Y. Thomas[349347]. Here, only symmetric connections can appear.
  3. The idea of mapping a manifold at one point to a manifold at a neighbouring point is central (affine, conformal, projective mappings). The names of König [192] and Cartan [29302] are connected with this program.

In his assessment, Eisenhart [121Jump To The Next Citation Point] adds to this all the geometries whose metric is

“based upon an integral whose integrand is homogeneous of the first degree in the differentials. Developments of this theory have been made by Finsler, Berwald, Synge, and J. H. Taylor. In this geometry the paths are the shortest lines, and in that sense are a generalisation of geodesics. Affine properties of these spaces are obtained from a natural generalisation of the definition of Levi-Civita for Riemannian spaces.” ([121Jump To The Next Citation Point], p. V)

In fact, already in May 1921 Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [297Jump To The Next Citation Point296Jump To The Next Citation Point]. In the first he wrote:

“Motivated by relativity theory, differential geometry received a totally novel, simple and satisfying foundation; I just refer to G. Hessenberg’s ‘Vectorial foundation...’, Math. Ann. 78, 1917, S. 187–217 and H. Weyl, Raum–Zeit–Materie, 2. Section, Leipzig 1918 (3. Aufl. Berlin 1920) as well as ‘Reine Infinitesimalgeometrie’ etc.107. [...] In the present investigation all 18 different linear connections are listed and determined in an invariant manner. The most general connection is characterised by two fields of third degree, one tensor field of second degree, and a vector field [...].”108View original Quote ([297Jump To The Next Citation Point], p. 57)

The fields referred to are the torsion tensor Sk ij, the tensor of non-metricity Qk ij, the metric g ij, and the tensor k C ij which, in unified field theory, was rarely used. It arose because Schouten introduced different linear connections for tangent vectors and linear forms. He defined the covariant derivative of a 1-form not by the connection Lkij in Equation (13View Equation), but by

+ ∂-ωi j ωi= ∂xk − ′L ki ωj, (130 )
with ′L ⁄= L. In fact
k k k Cij := L ij − ′Lij. (131 )
In the first paper, Schouten had considered only the special case Ckij = Ci δkj109.

Furthermore, on p. 57 of [297Jump To The Next Citation Point] we read:

“The general connection for n = 4 at least theoretically opens the door for an extension of Weyl’s theory. For such an extension an invariant fixing of the connection is needed, because a physical phenomenon can correspond only to an invariant expression.”110View original Quote

Through footnote 5 on the same page we learn the pedagogical reason why Schouten did not use the ‘direct’ method [294336] in his presentation, but rather a coordinate dependent formalism111:

“As the results of the present investigation might be of interest for a wider circle of mathematicians, and also for a number of physicists [...].”112View original Quote

At the end of the first paper we can find a section “Eventual importance of the present investigation for physics” (p. 79–81) and the confirmation that during the proofreading Schouten received Eddington’s paper ([58], accepted 19 February 1921). Thus, while Einstein and Weyl influenced Eddington, Schouten apparently did his research without knowing of Eddington’s idea. Einstein, perhaps, got to know Schouten’s work only later through the German translation of Eddington’s book where it is mentioned ([60Jump To The Next Citation Point], p. 319), and to which he wrote an addendum, or, more directly, through Schouten’s book on the Ricci calculus, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, in the same famous yellow series of Springer Verlag [300Jump To The Next Citation Point]. On the other hand, Einstein’s papers following Eddington’s [77Jump To The Next Citation Point74Jump To The Next Citation Point] inspired Schouten to publish on a theory with vector torsion that tried to remedy a problem Einstein had noted in his papers, i.e., that no electromagnetic field could be present in regions of vanishing electric current density. According to Schouten

“[...] we see that the electromagnetic field only depends on the curl of the electric current vector, so that the difficulty arises that the electromagnetic field cannot exist in a place with vanishing current density. In the following pages will be shown that this difficulty disappears when the more general supposition is made that the original deplacement is not necessarily symmetrical.” ([300Jump To The Next Citation Point], p. 850)

Schouten criticised Einstein’s argument for using a symmetric connection113 as unfounded (cf. Equation (15View Equation)). He then restricted the generality of his approach; in modern parlance, he did allow for vector torsion only:

“We will not consider the most general case, but the semi-symmetric case in which the alternating part of the parameters has the form:

′ ν ′ ν ν ν 1∕2(Γ μ λ − Γ λ μ) = 1∕2 (Sλ δμ − Sμ δλ),
in which Sλ is a general covariant vector.” ([298Jump To The Next Citation Point], p. 851)

The affine connection ′ Γ can then be decomposed as follows:

′ l l l Γ jk = Λjk + S [j δk]. (132 )
Hence, besides the covariant derivative ∇ ′ following from use of Γ ′ l jk, in his calculations Schouten also introduced a covariant derivative ∗ ∇ formed with l Λjk. Schouten’s point of departure for the field equations is Einstein’s first Lagrangian ∘ ------- ℒ = detKij and, consequently, his field equations were the same as Einstein’s apart from additional terms in vector torsion. Also, Schouten’s definition of some of the observables is different; For example, the electromagnetic field tensor unlike in Equation (125View Equation) is now
( ) ( ˆ ˆ ) ˆFkl = 1- ∂ˆık-− -∂ˆıl- − ∂-Sk − ∂-Sl , (133 ) 6 ∂xl ∂xk ∂xl ∂xk
where k ∗ kl kl i := ∇ lf − Plf, and ∂ ∘ --------- l Pk := − ∂xk(log − detKij ) + Λ lk. On the same topic, Schouten wrote a paper with Friedman in Leningrad [142Jump To The Next Citation Point]. A similar, but less detailed, classification of connections than Schouten’s has also been given by Cartan. He relied on the curvature, torsion and homothetic curvature 2-forms ([32Jump To The Next Citation Point], Section III; cf. also Section 2.1.4). In 1925, Eyraud came back to Schouten’s paper [298Jump To The Next Citation Point] and proved that his connection can be mapped projectively and conformally on a Riemannian space [124123]. Other mathematicians were also stimulated by Einstein’s use of differential geometry in his general relativity and, particularly, by the idea of unified field theory. Examples are Eisenhart and Veblen, both in Princeton, who developed the “geometry of paths”114 under the influence of papers by Weyl, Eddington, and Einstein [122117Jump To The Next Citation Point383]. In Eisenhart’s paper, we may read that

Einstein has said (in Meaning of Relativity) that ‘a theory of relativity in which the gravitational field and the electromagnetic field enter as an essential unity’ is desirable and recently has proposed such a theory.” ([117Jump To The Next Citation Point], pp. 367–368)


“His geometry also is included in the one now proposed and it may be that the latter, because of its greater generality and adaptability will serve better as the basis for the mathematical formulation of the results of physical experiments.” ([117], p. 369)

The spreading of knowledge about properties of differential geometric objects like connection and curvature took time, however, even in Leningrad. Seven years after Schouten’s classification of connections, Fréedericksz of Leningrad – known better for his contributions to the physics of liquid crystals – put forward a classification of his own by using both the connection and the curvature tensor [138].

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