André Lichnerowicz (; January 21, 1915,

Bourbon-l'Archambault Bourbon-l'Archambault () is a spa town and a commune in the Allier department in Auvergne-Rhône-Alpes region in central France. It is the place of origin of the House of Bourbon. Population Personalities In 1681, Louise Marie Anne de Bou ...

– December 11, 1998,

Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...

) was a French differential geometer and mathematical physicist. He is considered the founder of modern

Poisson geometry In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hami ...

Biography

His grandfather Jan fought in the Polish resistance against the

Prussia Prussia (; ; Old Prussian: ''Prūsija'') was a Germans, German state centred on the North European Plain that originated from the 1525 secularization of the Prussia (region), Prussian part of the State of the Teutonic Order. For centuries, ...

ns. Forced to flee Poland in 1860, he finally settled in France, where he married a woman from

Auvergne Auvergne (; ; or ) is a cultural region in central France. As of 2016 Auvergne is no longer an administrative division of France. It is generally regarded as conterminous with the land area of the historical Province of Auvergne, which was dis ...

, Justine Faure. Lichnerowicz's father, Jean, held

agrégation In France, the () is the most competitive and prestigious examination for civil service in the French public education A state school, public school, or government school is a primary school, primary or secondary school that educates all stu ...

in classics and was secretary of the

Alliance française (; "French Alliance", stylised as ''af'') is an international organization that aims to promote the French language and francophone culture around the world. Created in Paris on 21 July 1883 under the name ''Alliance française pour la propa ...

, while his mother, a descendant of paper makers, was one of the first women to earn the agrégation in mathematics. Lichnerowicz's paternal aunt, Jeanne, was a novelist and translator known under the pseudonym . André attended the

Lycée Louis-le-Grand The Lycée Louis-le-Grand (), also referred to simply as Louis-le-Grand or by its acronym LLG, is a public Lycée (French secondary school, also known as sixth form college) located on Rue Saint-Jacques (Paris), rue Saint-Jacques in central Par ...

and then the

École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i ...

, gaining agrégation in 1936. After two years, he entered the

Centre national de la recherche scientifique The French National Centre for Scientific Research (, , CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 staff, including 11,137 tenured researchers, 13,415 eng ...

(CNRS) as one of the first researchers recruited by this institution. Lichnerowicz studied

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

under

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

. His doctoral dissertation, completed in 1939 under the supervision of Georges Darmois, was entitled "''Problemes Globaux en Mécanique Relativiste''" (Global problems in relativistic mechanics). His academic career began under the cloud of

Nazi Nazism (), formally named National Socialism (NS; , ), is the far-right politics, far-right Totalitarianism, totalitarian socio-political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Germany. During H ...

occupation, during

World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

. In 1941 he started teaching at the

University of Strasbourg The University of Strasbourg (, Unistra) is a public research university located in Strasbourg, France, with over 52,000 students and 3,300 researchers. Founded in the 16th century by Johannes Sturm, it was a center of intellectual life during ...

, which was moved to Clermont Ferrand and only returned to

Strasbourg Strasbourg ( , ; ; ) is the Prefectures in France, prefecture and largest city of the Grand Est Regions of France, region of Geography of France, eastern France, in the historic region of Alsace. It is the prefecture of the Bas-Rhin Departmen ...

in 1945, after the end of the war. In November 1943 he was arrested during a raid but managed to escape. During 1944 he was invited to give a Cours Peccot at the

Collège de France The (), formerly known as the or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment () in France. It is located in Paris near La Sorbonne. The has been considered to be France's most ...

. From 1949 to 1952 he held a position at the

University of Paris The University of Paris (), known Metonymy, metonymically as the Sorbonne (), was the leading university in Paris, France, from 1150 to 1970, except for 1793–1806 during the French Revolution. Emerging around 1150 as a corporation associated wit ...

, and in 1952 he was appointed professor at the

, where he worked until his retirement in 1986. Lichnerowicz served as president of the

Société mathématique de France Groupe Lactalis S.A. (doing business as Lactalis) is a French multinational dairy products corporation, owned by the Besnier family and based in Laval, Mayenne, France. The company's former name was Besnier S.A. Lactalis is the largest dairy pr ...

during 1959. He was elected member of several national and international academies: the

Accademia dei Lincei The (; literally the "Academy of the Lynx-Eyed"), anglicised as the Lincean Academy, is one of the oldest and most prestigious European scientific institutions, located at the Palazzo Corsini on the Via della Lungara in Rome, Italy. Founded in ...

in 1962, the

Académie des Sciences The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...

in 1963, the Real Academia de Ciencias in 1968, the Académie Royale de Belgique in 1975, the

Pontifical Academy of Sciences The Pontifical Academy of Sciences (, ) is a Academy of sciences, scientific academy of the Vatican City, established in 1936 by Pope Pius XI. Its aim is to promote the progress of the mathematical, physical, and natural sciences and the study ...

in 1981, and the in 1984. In 1988 he was awarded the

Prix de la langue française The is chronologically the first grand prix of the literary season in France. Established in 1986 by the city of Brive-la-Gaillarde in the department of Corrèze, this prize rewards the work of a personality of the literary, artistic or scientific ...

for having illustrated the quality and the beauty of

French language French ( or ) is a Romance languages, Romance language of the Indo-European languages, Indo-European family. Like all other Romance languages, it descended from the Vulgar Latin of the Roman Empire. French evolved from Northern Old Gallo-R ...

in his works. In 2001 he received posthumous (together with his co-authors

Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...

and Marco Schutzenberger) the Peano Prize for his work ''Triangle of Thoughts.'' In 2008 the André Lichnerowicz Prize was created to reward progresses in

, a research field where Lichnerowicz made pioneering contributions. Lichnerowicz was a believing Catholic who served as vice-president of the Centre Catholique des Intellectuels Français.

Research

In an interview in his last years, Lichnerowicz self-described his research interests as "Differential geometry and global analysis on manifolds", "the relations between mathematics and physics" and "the mathematical treatment of Einstein’s theory of gravitation". Indeed, his works contributed, among others, to many areas of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

and

general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...

. His research in general relativity began with his PhD thesis, where he described necessary and sufficient conditions for a

metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...

of hyperbolic

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

to be a global solution of the

Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...

. In a series of papers in 1940 with Raymond Marrot, he provided a mathematical formulation of the relativistic kinetic theory. He later worked on

gravitational radiation Gravitational waves are oscillations of the gravitational field that travel through space at the speed of light; they are generated by the relative motion of gravitating masses. They were proposed by Oliver Heaviside in 1893 and then later by ...

, spinor fields, and propagators on curved space-time, obtaining results which preluded his later works on quantisation and deformation. Among his contributions to Riemannian geometry, in 1944 he formulated a conjecture about locally harmonic 4-manifolds, which has been later generalised and is now known as Lichnerowicz conjecture. In 1952 he showed, together with

Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...

, that the restricted

holonomy group In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...

of a

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

. He proved the now standard equivalence of the various definitions of

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

and he worked on the classification of compact

homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...

Kähler spaces. In 1958 he was one of the first to introduce a relation between the

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

of the

Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...

and the

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

of the metric. After formalising Cartan’s and

Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...

’s theory of

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

s in a rigorous framework, he proved in 1963 the Lichnerowicz formula relating the

Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...

and the

Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...

acting on spinors. In the 1970s his interests turned to symplectic geometry and dynamical systems, with many pioneering papers which, in the next decades, would give rise to the modern field of

. Indeed, starting in 1974, together with Moshé Flato and Daniel Sternheimer, Lichnerowicz formulated the first definitions of a Poisson manifold in terms of a

bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. Considering a scalar as a degree-zero quantity and a vector as a degree-one quantity, a bivector is of ...

, the counterpart of a (symplectic) differential 2-form. He showed later that the same philosophy can be used to generalise

contact structure In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...

s to Jacobi manifolds. In a 1976 paper one can already find the classical formula

f,dg d\

for the

Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...

bracket of

T^*M

on exact 1-forms via the

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

of functions. In 1977 Lichnerowicz introduced the operator defining what is now called Poisson cohomology. His 1978 papers on the deformation of the algebra of smooth functions on a Poisson manifold established the new research area of deformation quantisation. Lichnerowicz published more than 350 papers and supervised 24 Ph.D. students. A collection of scientific contributions from several of his collaborators was published in his honour in occasion of his 60th birthday. In 1982 a personal selection of his own works was published by Hermann.

Pedagogy of mathematics

While pursuing an active research career, Lichnerowicz had a deep interest in

mathematics education In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...

and

pedagogy Pedagogy (), most commonly understood as the approach to teaching, is the theory and practice of learning, and how this process influences, and is influenced by, the social, political, and psychological development of learners. Pedagogy, taken ...

. From 1963 to 1966 he was President of the International Commission on Mathematical Instruction of the

International Mathematical Union The International Mathematical Union (IMU) is an international organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports the International ...

. In 1967 the French government created the ''Lichnerowicz Commission'' made up of 18 teachers of mathematics. The commission recommended a curriculum based on

set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...

and

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

with an early introduction to

mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...

s. It recommended introduction to

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s for seniors in high school, less computation-based instruction, and more development from premises (the axiomatic approach). These reforms have been called New Math and have been repeated internationally. However, the reforms faced stern backlash from parents, who had trouble helping their children with homework, teachers, who found themselves ill-prepared and ill-equipped, and scholars from various disciplines, who deemed the New Math to be simply unsuitable and impractical. Lichnerowicz resigned and the commission was disbanded in 1973. Nevertheless, the influence of the proposed reforms in mathematics education had endured, as the Soviet mathematician

Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ...

recalled in a 1995 interview.

Works in French

* ''Problèmes globaux en mécanique relativiste'', Paris, Hermann, 1939. * ''Éléments de calcul tensoriel'', Armand Colin 1946, Éditions Jacques Gabay, 1987. * ''Algèbre et analyse linéaires, Paris, Masson'', 1947. * ''Les Théories relativistes de la gravitation et de l'électromagnétisme'', Paris, Masson, 1954. * ''Théorie globale des connexions et des groupes d'holonomie'', Rome, Cremonese, 1955. * ''Géométrie des groupes de transformations'', Paris, Dunod, 1958. * ''Propagateurs et commutateurs en Relativité générale'', Paris, PUF, 1961.

Works in English translation

* ''Elements of Tensor Calculus'', John Wiley and Sons, 1962
2016 Dover reprint
* ''Relativistic Hydrodynamics and Magnetohydrodynamics'', W. A. Benjamin, 1967. * ''Linear Algebra and Analysis'' Holden Day, 1967. (''Algèbre et analyse linéaires'', Paris, Masson, 1947) * ''Geometry of Groups of Transformations'', Leyden: Noordhoff,

958 Year 958 (Roman numerals, CMLVIII) was a common year starting on Friday of the Julian calendar. Events By place Byzantine Empire * October / November – Battle of Raban: The Byzantine Empire, Byzantines under John I Tzimiskes, Jo ...

1976. (''Géométrie des groupes de transformations'', Paris, Dunod, 1958) * ''Global Theory of Connection and Holonomy Groups'' Leyden: Noordhoff, 9551976. (''Théorie globale des connexions et des groupes d'holonomie'', Rome, Edizioni Cremonese, 1955), * ''Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time'' Kluwer, Springer 1994. * ''Chaos and Determinism'' (with Alexandre Favre, Henri Guitton and Jean Guitton), Johns Hopkins, 1995. * ''Triangle of Thoughts'' (with Alain Connes and Marco Schutzenberger),

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, 2000.

Notes