Gregorio Ricci-Curbastro

How to use this course

Gregorio Ricci-Curbastro is a name most physicists meet as half of a word: the Ricci tensor, the Ricci scalar, Ricci-flat, Ricci flow. Behind it stands one of the quietest revolutions in mathematics. Between 1884 and 1900, working almost alone in Padua, Ricci turned the scattered formulas of Riemann, Christoffel and Lipschitz into a systematic calculus of quantities that transform correctly under any change of coordinates. He called it the absolute differential calculus. We call it tensor calculus. Fifteen years after his memoir with his student Tullio Levi-Civita, it became the language of general relativity.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Conventions follow the Riemann, Einstein and Levi-Civita courses on this site: repeated indices are summed, \(\Gamma^i_{jk}=\tfrac12g^{im}(\partial_jg_{mk}+\partial_kg_{mj}-\partial_mg_{jk})\), \(R^i{}_{jkl}=\partial_k\Gamma^i_{lj}-\partial_l\Gamma^i_{kj}+\Gamma^i_{km}\Gamma^m_{lj}-\Gamma^i_{lm}\Gamma^m_{kj}\), and the Ricci tensor is \(R_{jl}=R^i{}_{jil}\), so that the round unit sphere has positive Ricci curvature.

Timeline

Lugo, Rome, Bologna, Pisa, 1853–1877

Gregorio Ricci-Curbastro was born on 12 January 1853 in Lugo, a market town of the Romagna between Ravenna and Bologna, then part of the Papal States. The Ricci-Curbastro family were landowners with estates around Lugo and a long history in agriculture. His father, Antonio, was an engineer well known in the province of Ravenna; his mother was Livia Vecchi. Gregorio and his brother Domenico did not go to school. Private tutors taught them at home, and Gregorio finished his secondary studies at sixteen.

In 1869 he enrolled at the University of Rome, the papal university, to study mathematics and philosophy. The choice was overtaken by events. In September 1870 Italian troops entered Rome and the city became the capital of the Kingdom of Italy. After a single year Ricci returned to Lugo and spent two years at home.

Bologna and Pisa

He resumed his studies at Bologna in 1872–73 and then moved to the Scuola Normale Superiore in Pisa, the strongest mathematical school in Italy. Enrico Betti directed the mathematical programme, and Ricci followed the lectures of Ulisse Dini. He took his laurea in 1875 with a thesis on Lazarus Fuchs’s research on linear differential equations. The following year he wrote, for his teaching qualification, a paper generalizing Riemann’s problem for hypergeometric functions. Neither work was published. They show a student trained in analysis and function theory, the Pisan tradition that Betti had brought back from his friendship with Bernhard Riemann.

Munich and Pisa, 1877–1880

Ricci’s first publications, in 1877, were on physics: a series of articles on James Clerk Maxwell’s electrodynamics and on work of Rudolf Clausius, three of them in Nuovo Cimento, and a paper on Lagrange’s problem for systems of linear differential equations. The interest in mathematical physics never left him. It was the subject he would teach for forty-five years.

In 1877–78 a scholarship took him to the Technische Hochschule in Munich, where Felix Klein and Alexander Brill were lecturing. He admired Klein, but his biographers judge that the decisive influences came from his reading of Riemann, of Elwin Bruno Christoffel’s 1869 paper on the transformation of quadratic differential forms, and of the parallel work of Rudolf Lipschitz. These authors had shown that the geometry of a quadratic form \(ds^2=g_{ij}dx^idx^j\) is contained in certain combinations of its derivatives, but their methods were heavy and particular. Ricci set out to find the general principle.

In 1879 he returned to Pisa as Dini’s assistant. In 1880, aged 27, he was appointed professor of mathematical physics at the University of Padua.

Padua and the making of a calculus, 1880–1900

Ricci lived the rest of his life in Padua. In 1884 he married Bianca Bianchi Azzarani; they had two sons and a daughter. From 1891 he also taught algebra. In the same years he published steadily on a single programme. In 1884 came Principi di una teoria delle forme quadratiche differenziali, built on Christoffel’s paper, which classified quadratic forms by an integer he called their class (Lesson 04). In 1886 he introduced covariant differentiation. In 1888 he gave the first systematic account of covariant and contravariant derivatives in a Paduan volume marking the eighth centenary of the University of Bologna. Four papers between 1888 and 1892 developed the technique, ending with a French résumé in the Bulletin des sciences mathématiques. In 1893 he named it the absolute differential calculus: absolute because its results do not depend on the coordinates.

Reception

The work was respected and little used. Eugenio Beltrami, the leading Italian differential geometer, acknowledged its importance but remained reserved. Many readers saw only a formalism. The Bulletin editors later summed it up as essentially a calculus of differential covariants of a quadratic form, a description that was accurate and missed the point. Ricci’s insistence on avoiding geometric language, which had historical reasons, made the work look more algebraic than it was.

Levi-Civita and Klein

In 1890 Tullio Levi-Civita entered the university. He graduated in 1892 with a thesis on absolute invariants under Ricci and quickly became his collaborator. Around 1895 Felix Klein, editor of Mathematische Annalen, asked Ricci for a full exposition. Ricci and Levi-Civita wrote Méthodes de calcul différentiel absolu et leurs applications, 77 pages in French, which appeared in volume 54, dated 1901 and usually cited as 1900. It is the only paper Ricci signed with the short form of his name, “G. Ricci”. Lessons 06 and 07 describe its contents.

Quadratic differential forms

For the general reader. A map of the Earth distorts distances, and different maps distort them differently. The rule for measuring distance on a given map is a quadratic differential form. Two forms may look completely different and still describe the same surface seen through different maps. Christoffel showed in 1869 how to decide whether they do. Ricci started from his work and asked a geometric question: in how many extra dimensions must a curved space live in order to sit inside flat space?

The equivalence problem

Two forms \(g_{ij}(x)\,dx^idx^j\) and \(\bar g_{ab}(y)\,dy^ady^b\) in \(n\) variables are equivalent if there is a change of variables \(x=x(y)\) with \(\bar g_{ab}=g_{ij}\,\frac{\partial x^i}{\partial y^a}\frac{\partial x^j}{\partial y^b}\). Christoffel introduced the symbols \(\Gamma^i_{jk}\) and the four-index symbols, the components of the Riemann tensor \(R_{ijkl}\), and showed that the question reduces to algebra on these quantities and their derivatives.

Proof sketch. If \(R=0\), the system \(\partial_j v^i=-\Gamma^i_{jk}v^k\) for parallel fields is integrable, since its compatibility condition is \(R^i{}_{jkl}v^j=0\). Choose \(n\) solutions forming an orthonormal frame at the point; they remain orthonormal. Their dual 1-forms \(\theta^a=v^a{}_i\,dx^i\) are closed because \(\Gamma\) is symmetric, so locally \(\theta^a=dy^a\), and \(ds^2=\sum(dy^a)^2\). The converse holds because \(R\) is a tensor that vanishes in the coordinates \(y\).

The class of a form (1884)

Ricci called a form of \(n\) variables of class \(k\) if it can be written, locally, as the metric induced on an \(n\)-dimensional submanifold of Euclidean \(\mathbb R^{n+k}\), and not for any smaller \(k\). Flat forms have class 0. Forms of class 1 are hypersurfaces, and Ricci studied them through an \(n\)-dimensional generalization of the second fundamental form.

For \(n=2\) the Gauss equation only says \(\det b=K\det g\), so infinitely many \(b\) exist; every analytic surface metric has class at most 1 locally. In general, Maurice Janet (1926) and Élie Cartan (1927) proved that every analytic metric in \(n\) variables has class at most \(n(n-1)/2\).

Covariant differentiation

For the general reader. In curved coordinates, ordinary derivatives of a vector depend on the coordinates as well as on the vector. Ricci’s central invention, the covariant derivative, adds correction terms built from Christoffel’s symbols so that the derivative of a tensor is again a tensor. With it, every formula of calculus in flat Cartesian space has a version that is valid in any coordinates and in any curved space.

Definitions

For a contravariant vector \(v^i\) and a covariant vector \(w_i\), Ricci defined

and for a general tensor one term of the first kind for each upper index and one of the second kind for each lower index. Ricci wrote the derivative as an extra index after a comma or semicolon, \(w_{i;j}\), a notation still in use.

Proof sketch. Under \(x\mapsto\bar x\) the Christoffel symbols transform as

The inhomogeneous term cancels exactly the second derivatives that appear when \(\partial_jv^i\) is transformed, which proves (a). For (c), \(\partial_kg_{ij}=\Gamma_{i,kj}+\Gamma_{j,ki}\) where \(\Gamma_{m,jk}=g_{mi}\Gamma^i_{jk}\); this is the definition of \(\Gamma\) read backwards. Part (d) holds because \(\nabla_i\nabla_jf=\partial_i\partial_jf-\Gamma^k_{ij}\partial_kf\) and \(\Gamma\) is symmetric.

The Riemann tensor appears as the obstruction to commuting covariant derivatives. This is the cleanest route to Christoffel’s four-index symbols, and it shows at once that they form a tensor.

The 1900 memoir: the algorithm

For the general reader. The memoir Ricci wrote with Levi-Civita for Klein’s journal is the founding document of tensor calculus. Its first chapter is a manual: how to add, multiply and contract tensors, how to differentiate them, and how to tell whether a formula is independent of coordinates. The remaining chapters show the method at work in geometry, analysis, mechanics and physics. Its message was that a well-chosen notation can do the thinking.

Plan of the memoir

The memoir opens with the algorithm of the absolute calculus: covariant and contravariant systems, the operations on them, covariant differentiation, and the Riemann tensor. It continues with intrinsic geometry used as a computing tool, through orthogonal frames attached to the manifold (Lesson 08). Applications follow: analytic (differential invariants and the classification of quadratic forms), geometric (surfaces and groups of motions), mechanical (Lagrange’s equations and dynamics) and physical (the equations of mathematical physics in arbitrary coordinates). The introduction credits the starting point of the algorithm to a remark of Christoffel.

The principle

The proof is the linearity of the transformation law. The working rule follows: to prove an identity, choose the coordinates in which it is easiest, for example coordinates in which \(g_{ij}=\delta_{ij}\) and \(\Gamma^i_{jk}=0\) at one point, prove it there, and observe that both sides are tensors. Ricci and Levi-Civita used such geodesic coordinates freely. A deeper statement, which goes back to Christoffel and which Tracy Thomas proved in general in the 1930s, says that every tensor built from \(g_{ij}\) and its partial derivatives can be expressed through \(g_{ij}\), the curvature tensor and its covariant derivatives.

Differential parameters

Gabriel Lamé and Eugenio Beltrami had studied differential parameters, expressions in the derivatives of a function that do not depend on coordinates. In the absolute calculus they become contractions:

The second is the Laplace–Beltrami operator. The equality of the two forms of \(\Delta_2\) follows from Exercise 5.1 applied to \(v^i=g^{ij}\partial_jf\).

The 1900 memoir: mechanics and physics

For the general reader. Ricci was a professor of mathematical physics, and the memoir’s last chapters show why the calculus mattered to him. Lagrange’s equations of motion, the equations of heat, of elasticity and of electromagnetism can all be written once, in tensor form, and then used in any coordinates. The motion of a system with constraints becomes geometry: free motion follows the straightest lines of a curved space defined by the kinetic energy.

Lagrange’s equations

Let a system have coordinates \(q^1,\dots,q^n\), kinetic energy \(T=\tfrac12g_{ij}(q)\dot q^i\dot q^j\) and generalized forces \(Q_i\). Lagrange’s equations \(\frac{d}{dt}\frac{\partial T}{\partial\dot q^i}-\frac{\partial T}{\partial q^i}=Q_i\) become

Proof. \(\frac{d}{dt}(g_{ij}\dot q^j)=g_{ij}\ddot q^j+\partial_kg_{ij}\dot q^k\dot q^j\) and \(\partial_iT=\tfrac12\partial_ig_{kl}\dot q^k\dot q^l\). The difference is \(g_{ij}\ddot q^j+\big(\partial_kg_{ij}-\tfrac12\partial_ig_{kj}\big)\dot q^k\dot q^j\). Symmetrizing in \(k,j\) turns the bracket into \(\Gamma_{i,kj}\), which gives the result. Constant speed follows from \(\frac{d}{dt}\big(g_{ij}\dot q^i\dot q^j\big)=2g_{ij}\dot q^i\frac{D\dot q^j}{dt}=0\).

Field equations

The same method rewrites the equations of mathematical physics. The heat equation \(\partial_tu=\kappa\,\Delta_2u\), Maxwell’s equations and the equations of elasticity take one tensor form, valid in every curvilinear system. In elasticity the strain tensor is \(e_{ij}=\tfrac12(\nabla_iu_j+\nabla_ju_i)\), and the compatibility conditions, which say that a strain comes from a displacement in flat space, are the linearized form of a vanishing curvature condition.

Orthogonal frames and rotation coefficients

For the general reader. Instead of coordinates, one can describe a curved space by attaching at each point a set of perpendicular unit arrows, like a surveyor’s cross carried around a field. Ricci measured how these arrows turn as one moves, with numbers now called the Ricci rotation coefficients. He used them in his 1896 memoir on congruences, families of curves filling a space, and they are still the standard tool for working with frames in relativity.

Definition

Let \(e_1,\dots,e_n\) be an orthonormal frame field, with components \(\lambda_h{}^i\), and dual covectors \(\lambda_{h|i}\). Each \(e_h\) is tangent to a congruence of curves. The rotation coefficients are

Since \(\langle e_h,e_k\rangle=\delta_{hk}\) is constant, \(\gamma_{hkl}=-\gamma_{khl}\). There are \(n^2(n-1)/2\) of them, and they contain the same information as the Christoffel symbols in the frame.

Proof sketch. (a) \(\nabla_{e_h}e_h=\sum_k\gamma_{hkh}e_k\) is the curvature vector of the curves. (b) The distribution orthogonal to \(e_h\) is integrable (Frobenius) if and only if \(\langle[e_k,e_l],e_h\rangle=0\) for \(k,l\ne h\). Since the connection is torsion-free, \(\langle[e_k,e_l],e_h\rangle=\langle\nabla_{e_k}e_l-\nabla_{e_l}e_k,e_h\rangle=\gamma_{lhk}-\gamma_{khl}=\gamma_{hkl}-\gamma_{hlk}\) by antisymmetry.

In 1896 Ricci used these tools to attach to any congruence a canonical orthogonal system and to study congruences in an arbitrary Riemannian manifold. In four-dimensional relativity the same coefficients, under the names spin coefficients and connection 1-forms, are the basis of the Newman–Penrose and tetrad formalisms.

The Ricci tensor and principal directions

For the general reader. The full curvature of a space is a complicated object with many components. Ricci averaged it in one natural way and obtained a simpler object, symmetric like the metric itself, which tells how volumes change as you move along a direction. He was led to it by asking which directions in a curved space are special, generalizing the directions of greatest and least bending on a surface. Fifteen years later it became the left-hand side of Einstein’s equations.

Definition

The Ricci tensor is the contraction \(R_{jl}=R^i{}_{jil}\). It is symmetric, by the first Bianchi identity. Its trace \(R=g^{jl}R_{jl}\) is the scalar curvature. For a unit vector \(v\), \(R_{jl}v^jv^l\) is the sum of the sectional curvatures of the \(n-1\) planes spanned by \(v\) and an orthonormal basis of \(v^\perp\). In normal coordinates the volume density is \(\sqrt{|g|}=1-\tfrac16R_{ij}x^ix^j+O(|x|^3)\), which is the picture at the top of the page.

The tensor appears in Ricci’s 1896 memoir on congruences. In Direzioni e invarianti principali in una varietà qualunque (1903–04) he studied its eigenvectors with respect to \(g\), the solutions of \((R_{ij}-\rho\,g_{ij})v^j=0\). He called them the principal directions of the manifold, and the eigenvalues \(\rho\) its principal invariants.

Proof. Contract the Gauss equation \(R_{ijkl}=b_{ik}b_{jl}-b_{il}b_{jk}\) over \(i,k\): \(R_{jl}=Hb_{jl}-b_{jk}b^k{}_l\). In the principal basis \(b=\mathrm{diag}(\kappa_i)\), so \(R_{jj}=H\kappa_j-\kappa_j^2\) and the off-diagonal entries vanish.

Proof sketch. Both sides are algebraic curvature tensors. Contracting the right side over \(i,k\) gives \((n-2)P_{jl}+g_{jl}\,\mathrm{tr}P=P_{jl}+\tfrac R4g_{jl}=R_{jl}\). In dimension 3 the space of algebraic curvature tensors and the space of symmetric 2-tensors both have dimension 6, and contraction is injective there, so the two sides agree. In dimension \(n\ge4\) the difference is the Weyl tensor (see the Weyl course).

From Ricci to Einstein

For the general reader. In 1912 Einstein knew that gravity had to be described by the metric of spacetime, but he did not know how to write equations for it that would look the same in every coordinate system. His friend Marcel Grossmann found the tools in the work of Riemann, Christoffel, Ricci and Levi-Civita. After three years of false starts, Einstein’s final equations of November 1915 set the Ricci tensor, corrected by its trace, equal to the energy and momentum of matter.

Grossmann and the Zurich notebook

In the summer of 1912 Einstein moved to the ETH and asked Grossmann, now his colleague, for help with the geometry of a variable metric. Grossmann pointed to the absolute differential calculus. In Einstein’s Zurich notebook of 1912–13 the Ricci tensor appears as a candidate for the gravitational field equations. Einstein set it aside because he could not reconcile it with the Newtonian limit and energy conservation as he then understood them, and the two men published the restricted Entwurf theory in 1913. In November 1915 he returned to the Ricci tensor and reached the final equations (see the Einstein course for the full story, and the Levi-Civita course for the 1915 correspondence).

Proof. Contract the second Bianchi identity over \(i=k\): \(\nabla_mR_{jl}+\nabla_iR^i{}_{jlm}-\nabla_lR_{jm}=0\), using \(R^i{}_{jmi}=-R_{jm}\). Contract again with \(g^{jl}\). The first term gives \(\nabla_mR\), the last \(-\nabla^jR_{jm}\), and the middle term \(\nabla_ig^{jl}R^i{}_{jlm}=-\nabla_iR^i{}_m=-\nabla^jR_{jm}\) by the pair symmetry of the Riemann tensor. So \(\nabla_mR-2\nabla^jR_{jm}=0\).

The energy–momentum tensor of matter satisfies \(\nabla^jT_{jl}=0\). Equations of the form \(G_{jl}=\kappa T_{jl}\) with \(G_{jl}=R_{jl}-\tfrac12Rg_{jl}\), the Einstein tensor, respect this automatically. Élie Cartan showed in 1922 that the only symmetric tensors built from the metric and its first two derivatives, linear in the second derivatives and divergence-free, are \(aG_{jl}+bg_{jl}\). David Lovelock proved in 1971–72 that in four dimensions the linearity assumption can be dropped.

Professor and citizen, 1900–1925

Ricci never left Padua. He taught mathematical physics there for forty-five years and algebra from 1891, and he wrote textbooks from his courses: Lezioni sulla teoria delle superficie (1898), Lezioni di algebra complementare (1900) and lectures on algebraic and infinitesimal analysis (1918), with a posthumous edition in 1926. He also wrote on the foundations of analysis, extending Richard Dedekind’s theory of real numbers in papers of 1893 and 1897.

He took civic duties seriously. As a councillor in Lugo he worked on the town’s water supply and on draining marshland, and drew up projects for land drainage in the province of Ravenna. As a councillor in Padua he dealt with schools and finance. He declined an offer to become mayor of Padua.

Recognition

Academic honours came steadily, though late relative to the importance of his work. He joined the Istituto Veneto in 1892 and presided over it from 1916 to 1918. He became a corresponding member of the Accademia dei Lincei in 1899 and a national member in 1916. He joined the Academy of Padua (1905) and the Academy of Sciences of Turin (1918), was president of the Accademia Galileiana from 1920 to 1922, and entered the Società dei Quaranta (1921), the Academy of Bologna (1922) and the Pontifical Academy (1925).

General relativity changed his standing. After 1915 Einstein repeatedly acknowledged his debt to the absolute calculus. On 27 October 1921 Einstein lectured in the Aula Magna of the Palazzo del Bo, the main building of the University of Padua, where Ricci had taught for four decades.

Death and legacy

Ricci-Curbastro died in Bologna on 6 August 1925, aged 72, still holding his Padua chair. He was buried in the family tomb in the cemetery of Lugo with his wife and relatives. His headstone records that he was a professor at the University of Padua for 45 years. A plaque on the family house in Lugo credits him with giving science the absolute differential calculus.

Legacy

• Language: the index calculus of tensors, with covariant and contravariant indices, covariant derivatives and contractions, is called the Ricci calculus. Albert Einstein’s summation convention was added to it in 1916.
• Physics: the Ricci tensor is the geometric side of Einstein’s field equations; vacuum spacetimes are Ricci-flat.
• Geometry: Ricci curvature controls volume growth and geodesic focusing (Bishop–Gromov comparison, Myers’s theorem). Ricci-flat Kähler manifolds, the Calabi–Yau spaces, are central in string theory.
• Ricci flow: Richard Hamilton’s equation \(\partial_tg_{ij}=-2R_{ij}\) (1982), with Grigori Perelman’s work of 2002–03, proved the Poincaré and geometrization conjectures.
• Names: Ricci tensor, Ricci scalar, Ricci identity, Ricci rotation coefficients, Ricci decomposition; the asteroid 13642 Ricci.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gregorio Ricci-Curbastro, MacTutor History of Mathematics.
2. Gregorio Ricci-Curbastro, Wikipedia.
3. P. Speziali, Ricci-Curbastro, Gregorio, Dictionary of Scientific Biography (MacTutor copy).
4. S. De Marchi and G. Andreatta, Ricci tensors and wine in Lugo di Romagna and Padova, 2017.
5. Commemorazione di Gregorio Ricci-Curbastro nel primo centenario della nascita, Rend. Sem. Mat. Univ. Padova 23 (1954).

Mathematics

1. G. Ricci-Curbastro, Principi di una teoria delle forme quadratiche differenziali, 1884.
2. G. Ricci-Curbastro, Résumé de quelques travaux sur les systèmes variables de fonctions, Bull. Sci. Math. 16 (1892).
3. G. Ricci and T. Levi-Civita, Méthodes de calcul différentiel absolu et leurs applications, Math. Ann. 54 (1901), 125–201.
4. G. Ricci-Curbastro, Direzioni e invarianti principali in una varietà qualunque, 1903–04.
5. E. B. Christoffel, Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades, J. reine angew. Math. 70 (1869).
6. T. Levi-Civita, The Absolute Differential Calculus, Blackie, 1927; Dover reprint.
7. Archive for History of Exact Sciences (2024), study of the geometric origins of Ricci’s absolute differential calculus.
8. Ricci curvature, Wikipedia.