Tullio Levi-Civita

Tullio Levi-Civita was born in 1873 in Padua, completed his doctorate at the University of Padua in 1894 under Gregorio Ricci-Curbastro, and spent his career first at Padua and later at the University of Rome, where he was appointed in 1918. With Ricci-Curbastro he co-authored the 1900 paper *Méthodes de calcul différentiel absolu et leurs applications* in *Mathematische Annalen* (volume 54, pages 125–201) — the work that systematised the index calculus on manifolds and provided the language Einstein would learn from Marcel Grossmann in Zurich in 1912 and use in November 1915 to write the field equations of general relativity. Without the *calcolo differenziale assoluto* there would have been no covariant tensor formalism for Einstein to write his theory in.

His 1917 paper *Nozione di parallelismo in una varietà qualunque* (Rendiconti del Circolo Matematico di Palermo, volume 42, pages 173–204) introduced the affine connection and parallel transport intrinsically, without reference to any embedding of the manifold in a higher-dimensional space. The concept gave the Christoffel symbols their modern geometric interpretation as connection coefficients, and the unique torsion-free metric-compatible connection that emerges from any Riemannian metric — what the world now calls the Levi-Civita connection — bears his name. So does the totally antisymmetric Levi-Civita symbol ε^{ijk}, which appears throughout differential geometry, electromagnetism, and field theory whenever an orientation has to be tracked. His correspondence with Einstein on the mathematical foundations of general relativity spanned twenty years and is one of the most consequential mathematician-physicist exchanges in the modern record.

Levi-Civita made important contributions to analytical mechanics and the three-body problem throughout his career, and trained a generation of Italian and international students. In 1938, after the racial laws of Mussolini's regime were passed, he was dismissed from his Rome professorship — he was Jewish — and stripped of all academic honours. He died in Rome on 29 December 1941, isolated and impoverished, his last years marked by the Nazi occupation closing in around the city. The connection that bears his name remains the standard tool with which every working physicist computes geodesics, curvature, and the field equations.