# Lichnérowicz, André

# LICHNéROWICZ, ANDRé

(*b*. Bourbon l’Archambault, France, 21 January 1915; *d*. Paris, France, 11 December 1998),

*mathematics, differential geometry, general relativity*.

Lichnérowicz, “Lichné” for his friends, was born 21 January 1915, in Bourbon l’Archambault, a small town in the center of France. His parents were highly educated people. Both of them were teachers, his father in the humanities and his mother in mathematics.

**Career** . Lichnérowicz studied differential geometry under the direction of Élie Cartan at the University of Paris. The thesis he defended under the direction of Georges Darmois in 1939 exemplifies the scientific orientation he took throughout his life, namely, bringing global geometric considerations into general relativity with concepts and formulas that allowed him to draw significant physical consequences. Indeed, he remained an active researcher until the end of his life.

In 1941 Lichnérowicz took the position of “maître de conferences” in mechanics at the Faculté des Sciences in Strasbourg that, during World War II, was operating in Clermont-Ferrand in the middle of France. When the war ended, the university returned to Strasbourg. In 1949 he was elected to a professorial position at the University of Paris, and in 1952 he received the great honor of being named to a chair at the Collège de France in Paris which, in the tradition of that special and elitist institution, bore a name tailored after the holder of the chair, in Lichnérowicz’s case, mathematical physics. He remained a professor there until 1986, when he reached the normal age of retirement.

In his lectures at Collège de France, which by the rules of the Collège had to cover a new topic each year, Lichnérowicz embraced a very broad landscape in mathematics and theoretical physics. He took the courses he taught as a natural opportunity to present the results of his research or to test out the content of a new book.

Unlike many of his colleagues, he did not limit his activity to producing new scientific results, and involved himself in changing mentalities in the rather conservative university environment in France. In particular, he played a crucial role in the organization of conferences in Caen and Amiens, in 1956 and 1960 respectively, which were instrumental in the modernization of the French university system. From 1966 to 1973 he also chaired the Commission for the Reform of Mathematics Teaching in France, (Commission pour la réforme de l’enseignement des mathématiques en France) known unofficially as the Lichnérowicz Commission, which had a direct impact on the mathematics instruction. There are diverse opinions about the changes that came out of the commission’s work, but it cannot be denied that the panel stirred up communities, and it made it plain that the training of teachers could not be insulated from the new developments in mathematics.

**Research Interests** . Lichnérowicz’s interests covered a wide variety of areas in mathematics and theoretical physics. He was a very important actor in the movement that brought geometry and physics closer together at the end of the twentieth century. He did so through the production of many new results; the training of research students; and the publication of a number of influential books, survey articles, and lectures.

His mathematical contributions belong to modern differential geometry as it developed in the twentieth century, moving from local considerations to global ones. It eventually led to the formal emergence of the central notion of a differential manifold, already envisioned by Bernhard Riemann. The growing importance of the field was significantly furthered by the parallel development of a theory of systems of nonlinear partial differential equations. General relativity exemplifies this revolution, particularly if one considers the much more ambitious challenges relating to the field that in the early 2000s can be tackled; the first fairly general situation where a global solution of the Einstein equation could be found was dealt with in 1990. The formulation of the equation that made this great achievement possible used the so-called “conformal formulation” introduced by Lichnérowicz in the 1980s.

Lichnérowicz’s approach to doing mathematics was peculiar in that he was always working on some new calculation, and he retained the urge to do so up to the very end of his life. He had an exceptional ability to see geometric facts through formulas. In some sense he was the perfect antidote to the Bourbaki approach to mathematics: He was continuously motivated by physical considerations and relied heavily on explicit computations as crystallizations of geometric facts.

Another key feature of Lichnérowicz’s considerable scientific legacy was his fascinating ability to put himself ahead of fashion. On topics such as holonomy groups, transformation groups, harmonic maps, symplectic geometry, and deformations of algebras of observables, he made substantial contributions precisely when these topics were not considered of central importance or even, in some cases, were viewed as marginal.

Lichnérowicz contributed substantially to a number of key areas. One was global differential geometry. In that field the notion of holonomy group attached to a covariant derivative took some time to emerge. It was introduced by Élie Cartan but it was Armand Borel and Lichnérowicz in 1952, and later Marcel Berger and Jim Simons, who established the key theorems that made it possible for the theory to become important.

In addition, a number of Lichnérowicz’s works deal with spaces with transitive transformations groups, the so-called homogeneous spaces. The special category of symmetric spaces, identified by Élie Cartan in the 1920s as providing very important families of models for important mathematical and physical situations that are completely amenable to algebraic computations, appears often in Lichnérowicz’s articles.

Lichnérowicz also gave great attention to the development of complex geometry and to the study of holomorphic maps between complex spaces. Motivated by his interest in theoretical mechanics, he pushed forward central concepts regarding symplectic geometry and groups of symplectic automorphisms. He played a key role in the establishment of Poisson geometry, one of symplectic geometry’s far-reaching generalizations, as a natural framework for the study of physically relevant models. It is therefore not surprising that he played an important role in improving the understanding of the structure of Kählerian manifolds, a notion that is at a crossroad between complex and symplectic geometries.

The theory of harmonic maps is in the early 2000s an established subject. This nonlinear generalization of harmonic analysis exists in modern theoretical physics under the name of nonlinear σ-models. Very early on, already in the 1970s, Lichnérowicz drew attention to the subject and made an important study of its relation to holomorphic maps when the spaces involved are complex and more specifically Kählerian.

It may be that Lichnérowicz proved himself a pioneer in the most obvious way in his development of a global theory of spinors on Riemannian and Lorentzian manifolds. Very early on, indeed in 1963, just after the proof of the index index theorem was announced, he understood the topological consequences for a compact spin manifold of the existence of a Riemannian metric with positive scalar curvature, by establishing a key formula relating the square of the Dirac operator to the usual Laplace-Beltrami operator on spinor fields. This formula was actually known to Erwin Schrödinger, a fact that has been overlooked for many years. Lichnérowicz was a strong advocate for the development of a self-standing spinorial geometry. The fact that he devoted his last works to Killing spinors, which mathematically underpin the concept of supersymmetry, and related objects is a proof of that opinion.

Lichnérowicz’s contributions to physics are also numerous, but general relativity is the topic to which he came back over and over again. As already mentioned, he strengthened the links between this theory and the strictly mathematical facts relevant to it. This led him to the study of a number of special models, especially in relation to the propagation of relativistic waves. He played a crucial role in the development of relativistic magnetohydrodynamics, a theory that ties together in a complicated way a physical analysis of the concepts and the mathematical tools needed to get something out of them.

On several occasions, he came back to the basic question of developing the concepts that allow deforming classical mechanics into quantum mechanics. For example, he explored the interplay between such tools and the structure of the underlying space.

Lichnérowicz published more than 350 research articles (including his lecture notes and contributions to conference proceedings) and seven books, which have been very influential and translated into several foreign languages. He was a dedicated thesis adviser who took great care of his research students. They in return showed great loyalty towards him.

Like his parents, Lichnérowicz was a highly cultivated man, curious about many culture and philosophies.

## BIBLIOGRAPHY

### WORKS BY LICHNÉROWICZ

With Armand Borel. “Groupes d’holonomie des variétés riemanniennes.” *Comptes Rendus Hebdomadaires des Séances de l Académie des Sciences Paris* 234 (1952): 1835–1837.

“Propagateurs et commutateurs en relativité générale.” *Publications mathématiques de l’Institut des hautes études scientifiques* 10 (1961): 293–344.

“Spineurs harmoniques.” *Comptes Rendus Hebdomadaires des Séances de l Académie des Sciences Paris* 257 (1963): 7–9.

“Variétés kählériennes et première classe de Chern.” *Journal of Differential Geometry* 1 (1967): 195–223.

*Global Theory of Connections and Holonomy Groups*. Translated from the French and edited by Michael Cole. Leyden: Noordhoff International Publishing, 1976.

*Geometry of Groups of Transformations*. Translated from the French and edited by Michael Cole. Leyden: Noordhoff International Publishing, 1977

“Les variétés de Poisson et leurs algèbres de Lie associees.”

*Journal of Differential Geometry* 12 (1977): 253–300.

*Choix de oeuvres mathématiques* [Selected mathematical works]. Paris: Hermann 1982.

“Killing Spinors, Twistor-Spinors and Hijazi Inequality.” *Journal of Geometry and Physics* 5, no. 1 (1988): 1–18.

With Alexandre Favre, Henri Guitton, Jean Guitton, and Etienne Wolff. *Chaos and Determinism: Turbulence as a Paradigm for Complex Systems Converging toward Final States*. Translated from the 1988 French original by Bertram Eugene Schwarzbach. With a foreword by Julian C. R. Hunt. Baltimore, MD: Johns Hopkins University Press, 1995.

*Magnetohydrodynamics: Waves and Shock Waves in Curved Space-Time*. Mathematical Physics Studies, 14. Dordrecht: Kluwer Academic Publishers Group, 1994.

With Alain Connes, and Marcel Paul Schützenberger *Triangle of Thoughts*. Translated from the 2000 French original by Jennifer Gage. Providence, RI: American Mathematical Society, 2001.

### OTHER SOURCES

Berger, Marcel; Jean-Pierre Bourguignon, Yvonne Choquet-Bruhat, et al. “André Lichnerowicz (1915–1998).” *Notices of the American Mathematical Society* 46 (December 1999): 1387–1396.

Cahen, Michel, and Moshe Flato, eds. *Differential Geometry and Relativity. A Volume in Honour of André Lichnerowicz on His 60th Birthday. Mathematical Physics and Applied Mathematics*, Vol. 3. Dordrecht and Boston: Reidel Publishing Co., 1976.

*Jean-Pierre Bourguignon*

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**Lichnérowicz, André**