Rossana Tazzioli, From Differential Geometry to Relativity: Levi-Civita’s Lectures on the Absolute Differential Calculus, 1925–1928, EMS Press, 2025, 523 pp., 99€, ISBN 978-3-98547-092-1, DOI 10.4171/hem/14
Forthcoming in British Journal for the History of Mathematics, doi 10.1080/26375451.2026.2677295

Tullio Levi-Civita (1873–1941) had a long and distinguished career in mathematics, marked by an early collaboration with his former doctoral advisor Gregorio Ricci Curbastro on tensor calculus (1901), and his discovery in 1917 of a geometric interpretation of covariant differentiation known as parallel transport. Both of these formal contributions are known to have exercised scientific imagination and practice in differential geometry, general relativity (GR) and unified field theory (UFT) in the 1920s, a circumstance that has given rise over the years to a wealth of scientific and scholarly works, respecting, for the most part, the same tripartite division.

Such piecemeal historiography, if understandable in light of the inherent historical complexity and conceptual thickness of these fields, remains unsatisfactory, as in general it neglects to investigate fully those lines of research that intersect domains of endeavor. In contrast to this standard approach, Rossana Tazzioli has produced a monograph that coordinates in a unified and compelling narrative the history of tensor calculus in the 1920s, centered on a textbook that appeared in mid-decade, Levi-Civita’s The Absolute Differential Calculus (hereafter, ADC). The subtitle notwithstanding, the genre is less “history of the book” than scientific biography. Tazzioli presents her work as biographical, and illustrates her strategy with an optical metaphor: ADC serves as a “prism to shed light on important facets of Levi-Civita’s activity” (p. 4). There are, naturally, many aspects of Levi-Civita’s scientific life which have no direct relation to his textbook, for instance, his contributions to UFT, and the anti-Semitic persecution he suffered under the fascist regime in Italy. In her monograph, Tazzioli takes up such topics while maintaining a focus on ADC and its early reception.

How does this prismatic, book-centered approach play out in a narrative? Remarkably well, thanks to Tazzioli’s sure grasp of the history of differential geometry, vector and tensor methods, the institutional history of mathematical sciences in Italy, and knowledge of Levi-Civita’s extensive international network of academics and research students. The latter is informed by a rich trove of incoming correspondence, several volumes of which Tazzioli co-edited, while his personal ties are illustrated by numerous photographs from the Levi-Civita estate.

The adoption by Einstein of Ricci and Levi-Civita’s tensor calculus (1901) for the presentation of his field equations of general relativity in 1915 represents a turning point for the history of tensor calculus. A year later, Levi-Civita, inspired by Einstein’s theory, made a momentous discovery, whereby he introduced the notion of parallel directions in an arbitrary Riemannian manifold, and related this to the manifold’s Riemannian curvature. An early adopter of Levi-Civita’s view, Hermann Weyl offered an intrinsic definition of the parallel transport of vectors in a semi-Riemannian manifold, which gave rise, in turn, to the notion of an affinely-connected manifold. Weyl also noticed that he could generalize Riemannian geometry to a gauge geometry, in which length is a path-dependent quantity, and he drew upon his discovery to formulate a field theory seeking to unify gravitation and electromagnetism, an ambitious project that attracted a large following in the 1920s, but which Weyl himself shortly abandoned.

The confirmation of GR by the results of the British eclipse expedition of 1919 greatly increased the exposure of GR and tensor calculus. When the young Wolfgang Pauli was tasked by Sommerfeld with the entry on relativity for Felix Klein’s Encyklopädie der mathematischen Wissenschaften (1921), he briefly introduced tensor analysis and parallel transport, along with Weyl’s affinely-connected spaces. What soon became known as the “Levi-Civita connection” made its appearance here, along with a summary of the ground-breaking contributions from Levi-Civita and Weyl on a generalization of Fermat’s principle in curved spacetime, and static, axisymmetric solutions to Einstein’s vacuum field equations.

When Levi-Civita left Padua for Rome in 1918, he introduced lectures on tensor calculus and relativity theory for his students, including Enrico Persico, whose notes of lectures on tensor calculus were published in 1925. The English version, in a translation by Marjorie Long (1926), included a presentation of geodesic deviation, and added two chapters on classical and relativistic mechanics and optics, and general relativity, respectively. The Viennese physicist A. Duschek produced an abbreviated version in German (1928), based on the Italian and English editions.

Tazzioli offers a helpful comparison of ADC with other textbooks and characterizes the production and reception of all three editions by drawing on Levi-Civita’s correspondence, some of which is transcribed and annotated in the appendix. These letters take up a full fifth of the monograph, and throw light on a variety of topics. One from Max von Laue (8 Jan., 1927), for instance, congratulates Levi-Civita on the publication of the English edition of ADC, and alerts him to a paper by Einstein and Jakob Grommer. Presented two days earlier to the Berlin Academy of Science, the paper, Laue writes, “connects with work by you and Weyl” (p. 383). The Einstein-Grommer paper recalls solutions of Einstein’s vacuum field equations for the static, axisymmetric case, and argues that they fully determine the equations of motion, provided that matter is represented by intrinsic singularities.

As Tazzioli notes (p. 91), Levi-Civita later took up the two-body problem using an alternative approach based on the full field equations. Following an exchange with Einstein in Princeton on 7 October, 1936, Levi-Civita modified his matter tensor in order to account for pressure. On this basis, Levi-Civita affirmed – mistakenly, as it happens – that the trajectory of an unperturbed conservative binary star is approximately circular, and he identified a binary ($b$ Persei) that could show the effect to astronomers. The paper on binary motion represents one of the final points on Levi-Civita’s life-path from differential geometry to relativity, and was one of the last contributions he published under university affiliation. In 1938, the racial decrees of the Fascist regime put an end to his teaching career.

Levi-Civita’s engagement with unified field theory stems similarly from Einstein’s research. In 1928, Einstein published a UFT of gravitation and electromagnetism known as teleparallelism, or Fernparallelismus. He subsequently learned he had reproduced connection components obtained earlier by Weitzenböck, and that the space of teleparallelism corresponds to a special case of a space discovered by Élie Cartan, endowed with a “Euclidean” connection, or a linear connection on a real, differentiable manifold, that features Riemannian curvature and what Cartan called “torsion”. As for Levi-Civita, he reformulated Einstein’s new theory by dropping the notion of teleparallelism altogether, in favor of Ricci’s method of orthogonal congruences of lines. With Einstein’s help, Levi-Civita’s reformulation was published in the proceedings of the Berlin Academy of Science. A few months later, Einstein backed his election to corresponding member of the Berlin Academy. Tazzioli observes that few were persuaded by teleparallelism, and that Einstein decided to “shift his focus away from unified field theories after 1931” (p. 239). In fact, when Einstein dropped teleparallelism in 1931, he did so in favor of a semivector theory, before returning to a five-dimensional approach in 1938. With the help of a slew of mathematicians and physicists (including Bergmann and Pauli), Einstein continued to publish UFTs until the end of his life in 1955.

Levi-Civita’s UFT appeared to hold promise for those seeking to recover Dirac’s relativistic electron theory (1928), although Dirac’s $\psi$-quantities (later “spinors”) do not transform like the four-vectors of Minkowskian relativity, and require instead the introduction of a complex representation of the Lorentz group. Weyl and others saw an occasion to reuse his notion of gauge invariance by replacing the length gauge with an imaginary phase gauge acting on the $\psi$-quantities, although Weyl doubted the possibility of obtaining a GR theory of two-component spinors. When Levi-Civita tried his hand at such a theory and failed, he placed the blame not on the resources of tensor calculus, but on Dirac’s equations, which he felt had to “be abandoned”, if only in GR (p. 246).

In summary, From Differential Geometry to Relativity is a fascinating read, and an important contribution to a unified history of differential geometry, general relativity and unified field theory in the 1920s.