Differential Geometry’s High Tide

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 ]:

The generalisation of parallel transport in the sense of Levi-Civita and Weyl. Schouten is the leading figure in this approach [300 ].
The “geometry of paths” considering the lines of constant direction for a connection – with the proponents Veblen, Eisenhart [122 , 114 , 115 , 373 ], J. M. Thomas [348 ], and T. Y. Thomas [349 , 347 ]. Here, only symmetric connections can appear.
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 [29 , 302 ] are connected with this program.

In his assessment, Eisenhart [121 ] 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.” ([121 ], p. V)

In fact, already in May 1921 Jan Arnoldus Schouten in Delft had submitted two papers classifying all possible connections [297 , 296 ]. 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 [...].”¹⁰⁸ ([297 ], 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 (13 ), but by

with $′L ⁄= L$ . In fact

In the first paper, Schouten had considered only the special case $Ckij = Ci δkj$ ¹⁰⁹ .

Furthermore, on p. 57 of [297 ] 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.”¹¹⁰

Through footnote 5 on the same page we learn the pedagogical reason why Schouten did not use the ‘direct’ method [294 , 336 ] 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 [...].”¹¹²

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 ([60 ], 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 [300 ]. On the other hand, Einstein’s papers following Eddington’s [77 , 74 ] 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.” ([300 ], p. 850)

Schouten criticised Einstein’s argument for using a symmetric connection¹¹³ as unfounded (cf. Equation (15 )). 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.” ([298

], p. 851)

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

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 (125

) is now

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 [142

]. 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 ([32

], Section III; cf. also Section 2.1.4). In 1925, Eyraud came back to Schouten’s paper [298

] and proved that his connection can be mapped projectively and conformally on a Riemannian space [124 , 123 ]. 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 [122 , 117

, 383 ]. 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.” ([117 ], pp. 367–368)

and

“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 ].