Tullio Levi-Civita

How to use this course

Every student of general relativity meets Tullio Levi-Civita twice in the first week: in the Levi-Civita connection, which tells a curved space how to differentiate, and in the Levi-Civita symbol, which writes determinants and cross products. Fewer know that he was above all a professor of rational mechanics, who worked on the three-body problem, on water waves and on vortex filaments, and who corrected Einstein’s tensor calculus by letter in 1915. This course presents that whole range alongside the story of a career that ended under the Fascist racial laws.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–07 form the geometric strand (1892–1926). Lessons 08–10 form the mechanical strand, which occupied him at least as much. Index conventions follow the Riemann and Einstein courses on this site: \(\Gamma^i_{jk}=\tfrac12g^{im}(\partial_jg_{mk}+\partial_kg_{mj}-\partial_mg_{jk})\) and \(R^i{}_{jkl}=\partial_k\Gamma^i_{lj}-\partial_l\Gamma^i_{kj}+\Gamma^i_{km}\Gamma^m_{lj}-\Gamma^i_{lm}\Gamma^m_{kj}\), with repeated indices summed.

Timeline

Padua, 1873–1892

Tullio Levi-Civita was born in Padua on 29 March 1873. His family was Jewish and well established in the city. His father, Giacomo Levi-Civita, was a lawyer who was appointed a senator of the Kingdom of Italy in 1908. The son grew up in a household where public service and the new Italian state were taken seriously.

He attended the classical secondary school in Padua and showed his mathematical talent early. In 1890 he entered the Faculty of Mathematics of the University of Padua. Two teachers shaped him. Giuseppe Veronese taught geometry in many dimensions. Gregorio Ricci-Curbastro, professor of mathematical physics, had spent the previous six years building a calculus for quadratic differential forms that hardly anyone outside Padua used.

The thesis

Levi-Civita graduated in 1892 with a thesis on absolute invariants written under Ricci-Curbastro. It was his first use of Ricci’s methods, and he already pushed them further, combining them with Sophus Lie’s theory of transformation groups. The work was printed in the proceedings of the Istituto Veneto. In 2016 the handwritten original, about a hundred pages, was found in the university archive in Padua.

The young professor, 1892–1900

After the laurea Levi-Civita took the teaching diploma (1894) and in 1895 began teaching at the teachers’ college attached to the Faculty of Science in Pavia. His output in these years was wide. In 1895 he improved Bernhard Riemann’s contour-integral formula for counting primes in an interval. In 1896 he published Sulle trasformazioni delle equazioni dinamiche, which asked when two dynamical systems have the same trajectories. The paper settled a problem Ulisse Dini had solved for surfaces, now in any dimension (Lesson 09).

In 1898, aged 25, he won the chair of rational mechanics at Padua. He would hold it for twenty years, and rational mechanics, the mathematical theory of motion, remained his subject for life. In 1899 he gave a group-theoretic interpretation of the integrals of a canonical system, linking first integrals of Hamiltonian systems to infinitesimal transformations. The note anticipates part of what Emmy Noether proved in general in 1918.

The memoir with Ricci

Felix Klein, editor of Mathematische Annalen, had asked Ricci-Curbastro for an exposition of his calculus. Ricci and his former student wrote it together. Méthodes de calcul différentiel absolu et leurs applications appeared in the volume dated 1901, pages 125–201, and is usually cited as the 1900 memoir. It set out the algebra and differential calculus of covariant and contravariant systems, now called tensors, and applied them to geometry, mechanics and mathematical physics. Lesson 04 and the Ricci-Curbastro course describe its content. The memoir was respected and little used for a decade. Then Marcel Grossmann showed it to Einstein.

The Paduan chair, 1900–1918

The first Padua decade of the century was dominated by mechanics. From 1903 Levi-Civita wrote a series of memoirs on the three-body problem, presented his qualitative study of the restricted problem at the International Congress in Heidelberg in 1904, and expanded it in Acta Mathematica in 1906. His aim was to remove the singularities caused by collisions (Lesson 08). In 1904 he gave conditions under which the Hamilton–Jacobi equation separates (Lesson 09). From 1906 he turned to hydrodynamics: the resistance of a fluid to a moving body, the wakes behind it, and with his student Luigi Sante Da Rios the motion of thin vortex filaments (Lesson 10).

Universities in Rome tried several times to attract him, including an effort by Guido Castelnuovo in 1909. He stayed in Padua. In 1914 he married Libera Trevisani, who had been one of his students and who later helped him with his papers.

Einstein and parallelism

In March 1915 Levi-Civita wrote to Einstein in Berlin. He had read Einstein’s 1914 account of the Entwurf theory and believed that its proof of the covariance properties of the gravitational field equations was wrong. Einstein defended the argument through several letters, then conceded in May, shortly before Italy entered the war. Lesson 07 tells the story. Two years later, reflecting on the new theory, Levi-Civita published the paper that defines parallel transport in an arbitrary Riemannian manifold. In 1918 he accepted the chair of higher analysis at the University of Rome, and he moved there at the start of 1919.

Absolute calculus and the Levi-Civita symbol

For the general reader. A physical law should not depend on the coordinates used to write it. Ricci-Curbastro’s absolute differential calculus guarantees this: it works with quantities, now called tensors, whose components change in a fixed way when the coordinates change, so an equation true in one system is true in all. Levi-Civita learned the method as a student, co-wrote its standard account, and taught it to physicists. One small piece of its notation, the alternating symbol, carries his name in every textbook.

Tensors

Let \(x^i\) and \(\bar x^{i}\) be two coordinate systems on an \(n\)-dimensional manifold. A system \(A_{i_1\dots i_r}{}^{j_1\dots j_s}\) is a tensor, covariant of rank \(r\) and contravariant of rank \(s\), if

Sums, products and contractions of tensors are tensors. The metric \(g_{ij}\), with \(ds^2=g_{ij}dx^idx^j\), is a covariant tensor of rank 2, and its inverse \(g^{ij}\) is contravariant. Raising and lowering indices with them identifies the two kinds, which is why the 1900 memoir could treat a vector field and its dual 1-form as one object.

The alternating symbol

The Levi-Civita symbol \(\varepsilon_{i_1\dots i_n}\) equals \(+1\) if \((i_1,\dots,i_n)\) is an even permutation of \((1,\dots,n)\), \(-1\) if odd, and \(0\) if an index repeats. It encodes the determinant: \(\det(a^i{}_j)\,\varepsilon_{j_1\dots j_n}=\varepsilon_{i_1\dots i_n}a^{i_1}{}_{j_1}\cdots a^{i_n}{}_{j_n}\). Alternating systems of this kind appear in the 1900 memoir, where Ricci and Levi-Civita used them to write invariants and the equations of physics without choosing coordinates.

Proof sketch. Both sides of the first identity are antisymmetric in \(j,k\) and in \(l,m\), and vanish unless \(\{j,k\}=\{l,m\}\). Check \(j=l=2,\ k=m=3\): the left side is \(\varepsilon_{123}^2=1\), the right side \(1\). Contract further for the other two.

Proof. With \(J=\partial x/\partial\bar x\), \(\bar g=J^{\mathsf T}gJ\), so \(\sqrt{|\bar g|}=|\det J|\sqrt{|g|}\). The determinant formula gives \(J^{i_1}{}_{k_1}\cdots J^{i_n}{}_{k_n}\varepsilon_{i_1\dots i_n}=\det J\;\varepsilon_{k_1\dots k_n}\). Multiplying, \(J\cdots J\,\eta=\det J\,\sqrt{|g|}\,\varepsilon=\sqrt{|\bar g|}\,\varepsilon=\bar\eta\) when \(\det J \gt 0\).

Parallel transport and the connection

For the general reader. On a flat sheet of paper you can carry an arrow from one point to another without turning it. On a curved surface there is no obvious meaning to “without turning”. In 1917 Levi-Civita gave one: move the arrow a tiny step, then discard the part that sticks out of the surface. The rule depends only on distances measured inside the surface, so it makes sense in any curved space, including spacetime. The idea of a connection, central to geometry and to gauge theories in physics, starts here.

Levi-Civita’s construction (1917)

The paper Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana appeared in the Rendiconti del Circolo Matematico di Palermo, vol. 42. Levi-Civita embedded a Riemannian manifold \(M^n\) in a Euclidean space \(\mathbb R^N\), which is possible locally. Let \(x(t)\) be a curve in \(M\) and \(V(t)\) a vector tangent to \(M\) at \(x(t)\). Call \(V\) parallel along the curve if its ordinary derivative in \(\mathbb R^N\) is normal to \(M\):

Write \(V=v^i\,\partial_iX\) where \(X(x^1,\dots,x^n)\) is the embedding. Then \(\frac{dV}{dt}=\dot v^i\,\partial_iX+v^i\dot x^j\,\partial_j\partial_iX\). Taking inner products with \(\partial_mX\) and using \(\partial_iX\cdot\partial_mX=g_{im}\) and \(\partial_j\partial_iX\cdot\partial_mX=\Gamma_{m,ij}\), the Christoffel symbol of the first kind, gives the condition

The embedding has disappeared. The equation depends only on \(g_{ij}\), so it defines parallel transport intrinsically, whatever the embedding. Levi-Civita also gave a direct characterization without embedding, using a variational argument modelled on virtual work in mechanics. Jan Arnoldus Schouten reached the same notion independently in 1918, and Hermann Weyl generalized it the same year to affine connections not derived from a metric.

The modern statement

A connection \(\nabla\) on \(M\) is metric if \(X\langle Y,Z\rangle=\langle\nabla_XY,Z\rangle+\langle Y,\nabla_XZ\rangle\) and torsion-free if \(\nabla_XY-\nabla_YX=[X,Y]\).

Proof. Write the metric condition three times with \(X,Y,Z\) permuted cyclically, add the first two, subtract the third, and use the torsion-free condition to replace differences \(\nabla_XY-\nabla_YX\). The result is Koszul’s formula

The right side does not involve \(\nabla\), so \(\nabla\) is unique; conversely the formula defines a connection with both properties. For coordinate fields the brackets vanish, and the formula is Christoffel’s expression. Preservation of inner products follows from \(\frac{d}{dt}\langle V,W\rangle=\langle\nabla_{\dot x}V,W\rangle+\langle V,\nabla_{\dot x}W\rangle=0\).

Curvature, holonomy, geodesic deviation

For the general reader. The second half of the title of the 1917 paper promised a “geometric specification of Riemannian curvature”. Carry an arrow around a small loop and it returns turned; the turning per unit area is the curvature. Levi-Civita returned to curvature in 1926, describing how nearby geodesics, the straightest paths, drift apart or come together. Physicists know the result as the equation of tidal forces.

Curvature as a commutator

Covariant derivatives do not commute. With the conventions of Lesson 00, for any vector field \(v\),

This is the Ricci identity of the absolute calculus. A space is flat exactly when every vector can be transported independently of path, which is the condition \(R^i{}_{jkl}=0\).

For a surface this sharpens to an exact statement. If \(\Delta\) is a geodesic triangle with angles \(\alpha,\beta,\gamma\), a vector transported around its boundary returns rotated by \(\alpha+\beta+\gamma-\pi=\iint_\Delta K\,dA\), which is Gauss’s theorema elegantissimum read through parallel transport. The figure at the top of this page shows the case of an octant of the sphere.

The parallelogramoid

To make transport concrete without equations, Levi-Civita built small quadrilaterals whose sides are geodesic segments, generalizing the Euclidean parallelogram. In flat space the figure closes exactly. In a curved space it fails to close at second order in its size, and the defect is governed by the curvature. The construction, the Levi-Civita parallelogramoid, is a cousin of Schild’s ladder, which approximates parallel transport by geodesic midpoints and is used in relativity and in numerical geometry.

Geodesic deviation

Let \(\gamma_s(t)\) be a family of geodesics and \(J=\partial_s\gamma\) the separation vector, with \(T=\partial_t\gamma\). Since \([T,J]=0\) and \(\nabla_TT=0\), \(\nabla_T\nabla_TJ=\nabla_T\nabla_JT=\nabla_J\nabla_TT+R(T,J)T=R(T,J)T\). In components,

This is the equation of Carl Jacobi for surfaces, extended to any dimension; Levi-Civita gave the general form in a 1926 paper on geodesic deviation (écart géodésique). In general relativity it describes tidal acceleration between freely falling particles.

Einstein and general relativity

For the general reader. In 1915 Einstein was struggling to find field equations for gravity that would hold in every coordinate system. A professor in Padua wrote to point out that one of his proofs was wrong. Einstein resisted, then agreed. The exchange did not give Einstein his final equations, but it showed him that the Italian school understood his mathematics better than he did, and it began a long friendship.

The 1915 letters

In 1913 Einstein and Grossmann had published the Entwurf, a theory in which the metric is the gravitational field but whose field equations are covariant only under a restricted class of “adapted” coordinate changes. In a long 1914 paper in the proceedings of the Prussian Academy, Einstein claimed to derive this restriction from a variational principle and gave a proof that his gravitational tensor transformed correctly within the adapted class. Levi-Civita’s first letter, in March 1915, argued that the proof did not work. Einstein replied at length, defending it. Letters and postcards went back and forth until May, when Einstein accepted that his argument had a gap. Levi-Civita kept Einstein’s replies; historians reconstructed both sides of the exchange from his archive. By November 1915 Einstein had abandoned the Entwurf and reached generally covariant field equations (see the Einstein course). Writing to Levi-Civita he praised the elegance of the Italian’s method of calculation.

After 1915 the two men also discussed gravitational energy and gravitational waves. Levi-Civita argued for treating the Einstein tensor itself, up to a constant, as the energy tensor of the gravitational field; Einstein preferred a non-covariant pseudotensor. The question of local gravitational energy, which also occupied David Hilbert, Felix Klein and Emmy Noether, has no covariant answer, as Noether’s second theorem explains.

Static fields

Between 1917 and 1919 Levi-Civita published a series of notes in the Rendiconti of the Lincei on Einstein metrics in Newtonian fields, solving the field equations for static configurations. Several vacuum solutions found there, including a static cylindrically symmetric family, are still called Levi-Civita metrics. The basic equations are these.

The first equation is the relativistic counterpart of Laplace’s equation for the Newtonian potential: with \(V\approx1+\Phi\) and \(h\approx\delta\), it reduces to \(\Delta\Phi=0\).

The three-body problem and regularization

For the general reader. Newton’s equations for planets break down when two bodies collide: the force becomes infinite and the solution stops. Levi-Civita found a change of variables, squaring a complex number and slowing down time near the collision, that turns the dangerous motion into the gentle swing of a pendulum. The colliding bodies simply bounce off each other in the new variables, and the solution continues. His trick is still used in every serious code that integrates close encounters of stars and planets.

Collisions

In the three-body problem, Paul Painlevé showed in 1897 that the only singularities are collisions. Collision orbits are exceptional, but they cannot be ignored when one wants statements valid for all orbits, and near-collisions make numerical integration fail. Levi-Civita’s papers of 1903–1906 on the restricted problem, where a massless body moves in the field of two primaries on circular orbits, aimed to show that a binary collision is not a true end of motion. His long 1920 memoir in Acta Mathematica collected the method.

The transformation

Write the plane as \(\mathbb C\) and consider the Kepler problem \(\ddot z=-\mu z/|z|^3\) with energy \(h=\tfrac12|\dot z|^2-\mu/|z|\). Introduce

Derivation. With \({}'=d/ds\), \(z'=|u|^2\dot z\) and \(z'=2uu'\), so \(\dot z=2u'/\bar u\). The energy becomes \(h=(2|u'|^2-\mu)/|u|^2\). Differentiate \(\dot z=2u'/\bar u\) with respect to \(s\): \(|u|^2\ddot z=2u''/\bar u-2u'\bar u'/\bar u^2\). Substituting \(\ddot z=-\mu u^2/|u|^6\) and multiplying by \(\bar u/2\) gives \(u''=|u'|^2u/|u|^2-\mu u/(2|u|^2)=\tfrac12\big(2|u'|^2-\mu\big)u/|u|^2=\tfrac{h}{2}u\). For bound orbits \(h \lt 0\) and \(u\) is a two-dimensional harmonic oscillator: an ellipse centred at the origin, whose square is a Kepler ellipse with focus at the origin.

Context

In 1912 Karl Sundman, by a different and indirect method that Levi-Civita acknowledged, proved that for nonzero total angular momentum the three-body problem has a solution expressible as a power series in a uniformizing variable convergent for all time. The series converge far too slowly to compute with. Levi-Civita’s map is the square map of the complex plane, and the time change removes the singular velocity. In 1964–65 Paul Kustaanheimo and Eduard Stiefel extended it to three dimensions by passing to four-dimensional \(u\) via quaternions; the KS transformation is used in modern N-body codes.

Analytical mechanics: separability

For the general reader. Most mechanical problems cannot be solved exactly. A few can, because in well-chosen coordinates they split into independent one-dimensional problems. Levi-Civita gave a test that decides from the energy function alone whether a given coordinate system achieves this splitting. Earlier he had asked a related question: when do two different mechanical systems trace out the same paths?

Same trajectories (1896)

A natural mechanical system with kinetic energy \(\tfrac12g_{ij}\dot q^i\dot q^j\) and no forces moves along geodesics of \(g\). Two metrics are geodesically equivalent if they have the same geodesics as unparametrized curves. Ulisse Dini had found all such pairs on surfaces in 1869. In his 1896 memoir Levi-Civita found local normal forms for all geodesically equivalent pairs in any dimension. The question returned in the 1990s through the integrable systems it produces.

Separation of variables (1904)

The Hamilton–Jacobi equation \(H(q,\partial S/\partial q)=E\) separates in the coordinates \(q^1,\dots,q^n\) if it has a complete solution \(S=\sum_iS_i(q^i;\alpha)\). Levi-Civita’s 1904 paper in Mathematische Annalen gave a criterion.

Idea of proof. In a separated solution, each momentum \(p_i=S_i'(q^i)\) depends on \(q^i\) and the constants only. Differentiating \(H(q,p(q))=E\) and requiring \(\partial p_i/\partial q^k=0\) for \(k\ne i\) gives \(\partial p_i/\partial q^i=-H_{q_i}/H_{p_i}\); the compatibility of these relations under a second differentiation is the stated system.

Hydrodynamics: wakes, vortices, waves

For the general reader. From 1906 Levi-Civita spent much of his energy on fluids. He studied the wake behind a body moving through water, guided a student to the equation that governs a thin twisted vortex, and in 1925 proved that waves of finite height can travel across deep water forever without changing shape, settling a question raised by George Stokes in 1847.

Wakes

In the classical free-streamline theory of Helmholtz and Kirchhoff, a plate in a stream leaves behind a region of dead water bounded by surfaces of constant pressure. Levi-Civita’s memoirs from 1906 onward treated the general obstacle in two dimensions by conformal mapping, introducing an auxiliary complex function that reduces the free-boundary problem to a problem in a fixed domain. The method is still known by his name in the theory of cavities and jets.

Vortex filaments

In his 1906 thesis under Levi-Civita, Luigi Sante Da Rios showed that a thin vortex filament \(X(\sigma,t)\), parametrized by arc length, moves to leading order in the direction of its binormal with speed proportional to its curvature:

with a coefficient \(\Gamma_{\!\text{eff}}\) depending logarithmically on the ratio of core radius to curvature radius. Levi-Civita developed the asymptotics in later papers. The equation was forgotten and rediscovered in the 1960s as the local induction approximation. In 1972 Hidenori Hasimoto showed that \(\psi=\kappa\,e^{i\int\tau\,d\sigma}\), built from curvature \(\kappa\) and torsion \(\tau\), satisfies the cubic nonlinear Schrödinger equation, so vortex filaments carry solitons.

Permanent waves

Consider irrotational flow of an ideal fluid of infinite depth under gravity \(g\), with a free surface. Stokes computed periodic travelling waves as power series in the amplitude, finding speed \(c=\sqrt{g/k}\,(1+\tfrac12k^2a^2+\cdots)\) for wavenumber \(k\) and amplitude \(a\). Whether the series describe true solutions was open.

Levi-Civita mapped the unknown flow domain conformally to a fixed one, as in his wake work, and proved convergence of the resulting series by majorants. Aleksandr Nekrasov had reached the result independently in Russia in 1921 by an integral equation. Dirk Struik extended Levi-Civita’s proof to finite depth in 1926. Global branches up to the wave of greatest height, with its 120° crest, came much later.

Rome, 1918–1938

Levi-Civita began teaching at the University of Rome on 1 January 1919, first as professor of higher analysis and from 1920 in the chair of rational mechanics. Rome then had a strong mathematical faculty, with Vito Volterra, Guido Castelnuovo and Federigo Enriques, and the young Enrico Fermi arrived in 1926. Levi-Civita became the centre of a school in mechanics and differential geometry. His doctoral students in Padua and Rome included Antonio Signorini, Attilio Palatini, Libera Trevisani, Octav Onicescu, Gheorghe Vrânceanu, Albert Joseph McConnell, Evan Tom Davies and Giovanni Lampariello. Students came from Romania, Ireland, Britain and elsewhere.

His teaching produced books that circulated widely: Questioni di meccanica classica e relativistica (1924, in German the same year), Lezioni di calcolo differenziale assoluto (1925, edited from his lectures by Enrico Persico and translated into English in 1927 as The Absolute Differential Calculus), the two volumes of Lezioni di meccanica razionale with Ugo Amaldi (1926–27), and Fondamenti di meccanica relativistica (1928). He worked on adiabatic invariants, the motion of bodies of variable mass, and in 1933 on the Dirac equation.

International life

Levi-Civita was a convinced internationalist. In 1920 he corresponded with Arnold Sommerfeld about restoring scientific relations after the war. With Theodore von Kármán he initiated the 1922 Innsbruck meeting on hydro- and aerodynamics, from which grew the International Congresses of Applied Mechanics, first held at Delft in 1924; he served on their committee until his death. He received the Sylvester Medal of the Royal Society in 1922 and was elected a Foreign Member in 1930, the year he attended the Edinburgh Mathematical Society’s colloquium at St Andrews. He was a member of the Accademia dei Lincei and of the American Academy of Arts and Sciences (1917), and was named to the Pontifical Academy of Sciences.

The oath of 1931

In 1931 the Fascist government required every university professor to swear loyalty to the regime. Twelve of about 1,250 refused, among them Volterra. Levi-Civita opposed the oath on moral grounds but signed, for the sake of his family and his research group in Rome. He travelled widely in the following years: the United States in 1933, Moscow and Kiev in 1935, and in 1936 Harvard, Princeton and the Rice Institute in Houston. After a newspaper interview in Houston was read as critical of Italy, he was ordered home. Italian mathematicians were barred from the 1936 International Congress in Oslo, yet the Congress named him to the committee that awarded the first Fields Medals.

The racial laws, death and legacy, 1938–1941

In the autumn of 1938 the Fascist government enacted laws against Jews, beginning with a decree of 5 September that excluded people of Jewish descent from schools and universities. Levi-Civita, then 65 and the most internationally honoured Italian mathematician of his generation, was dismissed from his chair at Rome. He was removed from the Accademia dei Lincei and the other Italian academies, forbidden to use the mathematical library of his own university, and taken off the editorial board of the German Zentralblatt für Mathematik. He could not attend the International Congress of Applied Mechanics in the United States in 1939.

He stayed in Rome with his wife. Despite failing health, he used his correspondence to help younger colleagues and students who had also been expelled find positions abroad, many in South America and the United States. His heart condition worsened after the dismissal. He died of a stroke in his Rome apartment on 29 December 1941, aged 68. His last major work, Le problème des n corps en relativité générale, was published posthumously in 1950.

Legacy

• Geometry: parallel transport made the curvature of Riemann and Christoffel visible, and led directly to Weyl’s affine connections, Élie Cartan’s general connections and holonomy, and the connections on fibre bundles of modern geometry.
• Physics: the Levi-Civita connection is the gravitational field of general relativity; gauge fields are connections in the same sense. The tensor calculus he co-wrote with Ricci-Curbastro is the language of the theory.
• Mechanics: Levi-Civita regularization, the separability conditions, the Da Rios–Levi-Civita filament equation and the existence of Stokes waves remain in active use.
• Notation: the Levi-Civita symbol; also the Levi-Civita field, an ordered field containing infinitesimals, which goes back to his papers of the 1890s on actual infinitesimals.
• Memory: his collected Opere matematiche appeared from 1954. The mathematics department of the University of Padua is named after him, and so is a lunar crater.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Tullio Levi-Civita, MacTutor History of Mathematics.
2. Tullio Levi-Civita, Wikipedia.
3. E. T. Whittaker, Tullio Levi-Civita 1873–1941, Obituary Notices of Fellows of the Royal Society 4 (1942).
4. Levi-Civita, Tullio, Dictionary of Scientific Biography (MacTutor copy).
5. University of Padua, Tullio Levi-Civita, materials for the 2018 conference.
6. P. Nastasi and R. Tazzioli, Toward a scientific and personal biography of Tullio Levi-Civita, Historia Mathematica 32 (2005).

Mathematics and physics

1. G. Ricci and T. Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Math. Ann. 54 (1901), 125–201.
2. T. Levi-Civita, Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana, Rend. Circ. Mat. Palermo 42 (1917), 173–205.
3. T. Levi-Civita, Sur la résolution qualitative du problème restreint des trois corps, Acta Math. 30 (1906), 305–327.
4. T. Levi-Civita, Sur la régularisation du problème des trois corps, Acta Math. 42 (1920).
5. T. Levi-Civita, Détermination rigoureuse des ondes permanentes d’ampleur finie, Math. Ann. 93 (1925), 264–314.
6. T. Levi-Civita, The Absolute Differential Calculus, Blackie, 1927; Dover reprint.
7. G. Weinstein, Einstein the stubborn: correspondence between Einstein and Levi-Civita, arXiv:1202.4305.
8. G. Iurato and G. Ruta, On the role of virtual work in Levi-Civita’s parallel transport, Arch. Hist. Exact Sci. 70 (2016).
9. Levi-Civita connection, Wikipedia (with historical notes).
10. R. L. Ricca, Rediscovery of Da Rios equations, Nature 352 (1991).