This book examines Levi-Civita’s lectures on tensor calculus as a lens to illuminate key aspects of his scientific legacy. It highlights the deep interplay between his teaching and research, particularly in tensor calculus, differential geometry, and relativity, as well as his role as a mentor at the University of Rome. More broadly, it traces the history of Riemannian differential geometry from roughly 1870 to 1930.

Key themes emerge: the influence of the Italian mathematical tradition in Levi-Civita’s work on tensor calculus, the intrinsic link between analysis, geometry, and relativity in his work, and his pedagogical approach, which incorporates physics and geometric intuition to extend mathematical results. The book also explores his collaborations with Enrico Fermi and Enrico Persico, shedding light on the Via Panisperna group during a pivotal period in theoretical physics.

Levi-Civita’s treatise became a foundational text in absolute differential calculus, essential for physicists mastering tensor calculus in Einstein’s theories. Drawing extensively from his archives — preserved at the Archivio Storico dell’Accademia Nazionale dei Lincei in Rome and within the Ceccherini-Silberstein family – the book offers fresh insights into his personal, scientific, and academic life. His correspondence reveals his far-reaching influence, spanning students in Rome, international scholars, Rockefeller fellows, and colleagues inspired by his ideas and mentorship.