# Paulette Libermann

### Quick Info

Paris, France

Montrouge, near Paris, France

**Paulette Libermann**was a French mathematician who survived the terrors of World War II and made important contributions to differential geometry.

### Biography

**Paulette Libermann**was born into a Jewish family who were of Polish and Ukrainian origin. Perhaps 'Polish' is not strictly accurate since Poland did not exist at that time and so many refer to her background as Russian. Her maternal grandparents were Israël Mayer Apte (1851-1938), born in Warsaw, and Rachel Moïse Schoub (1852-1921) who were married in Paris on 1 April 1875. Her parents spoke both French and Yiddish at home, but Paulette and her two sisters were French speaking. Paulette attended the Lycée Lamartine, a school for girls in the rue du Faubourg, Poissonnière, Paris. There she excelled, showing her brilliant abilities, and often helping her fellow pupils with their homework. In 1938 she entered the École Normale Supérieure de Sèvres. This was a university level institution designed to educate women who were going to become secondary school teachers. It was also known as the École Normale Supérieure de Jeunes Filles, and the students there studied a special women's scientific teaching diploma. It had been founded in 1881 and remained a separate institution until it joined the École Normale Supérieure in 1987. However around the time Libermann was a student, the director was Eugénie Cotton (1881-1961), a physicist with strong left wing views, and she wanted to make the École Sèvres educate women to the same level as the École Normale Supérieure at the Rue d'Ulm trained men. Libermann greatly benefited from the reforms brought in by Eugénie Cotton, and she was taught by leading mathematicians such as Élie Cartan, André Lichnerowicz and Jacqueline Ferrand.

World War II began in September 1939 and in May 1940 [12]:-

... the German army invaded a large part of France that included Paris, and there followed in the spring and summer of 1940 a southward exodus of civilians. Paulette and her family took refuge in Lyons, where she was able to sit for two examinations. She then travelled to Toulouse to sit for additional examinations. She returned to Paris with her family in autumn. Part of the École had moved to Paris after the German troops occupied the building in Sèvres, and she prepared for the agrégation while living with her family.After the defeat of France in World War II, the Vichy Regime had taken over control of the country. It enacted laws to imitate those brought in by the Third Reich in Germany and, in particular, in October 1940 they passed a Statute excluding Jews from many occupations such as the armed forces, entertainment, arts, media, and professional jobs like the teaching profession. This made it impossible for Libermann to take the examinations for her teaching certificate, but Eugénie Cotton was able to obtain scholarships for three Jewish students to study at the École Sèvres for a fourth session. Eugénie Cotton was forced to retire in 1941, but Libermann, who was one of the three Jewish students supported by the scholarships, was able to spend session 1941-42 at École Sèvres beginning a research career advised by Élie Cartan [3]:-

With a kind of black humour, Paulette Libermann used to say that the anti-Semitic laws had been lucky for her, since Élie Cartan, who was teaching these young ladies, suggested that she start research instead [of becoming a teacher].From that time on, Libermann had great admiration for Élie Cartan and remained a close friend of the Cartan family for the rest of her life.

By the spring of 1942 the conditions imposed on Jews by the Vichy Regime was becoming increasingly severe. From February 1942 telephones and radios found in the homes of Jewish people had been confiscated and a curfew had been enforced on Jews. Libermann, with her parents and two sisters, fled from Paris in June 1942 after the Vichy Regime required Jews to wear a yellow star. They went to Lyon where they lived under false names, but conditions there soon became desperate. In November 1942 Klaus Barbie was sent to Lyon where he became the head of the local Gestapo. He is thought to have been responsible for the deaths of 4000 Jews in Lyon during the two years he served there and, as a result of this and the fact that he personally tortured those captured, he is known as the "Butcher of Lyon". Somehow Libermann and her family were able to hide from Barbie despite the fact they lived in la Place Bellecour which adjoined a building occupied by the Gestapo. In later life she described the fact that they survived this terrible period as a miracle. In the autumn of 1944, following the liberation of Paris by the Allies in August, Libermann was able to return to the École Sèvres in Paris and obtain her teaching certificate.

After the award of her certificate, Libermann was appointed to teach, first at Douai in the north east of France about 30 km south of Lille, then, from the beginning of the 1945, at the High School for Girls in Strasbourg. Again she received important advice from Élie Cartan who suggested to her she contact Charles Ehresmann at the University and begin studying for a doctorate in mathematics under his supervision. This she did while continuing to teach at the High School for Girls. To finish off writing her thesis, she spent a while at Centre National de la Recherche Scientifique. Then in 1953 she defended her thesis

*Sur le problème d'équivalence de certaines structures infinitésimales*Ⓣ and received her Docteur d'Etat from the Louis Pasteur University of Strasbourg. Audin writes [1]:-

The equivalence problem is a general problem and has been investigated by many including Élie Cartan. Roughly speaking, the question is to classify, up to local isomorphism, structures on manifolds. For instance, all the manifolds of the same dimension are equivalent (this is a local question). But this is not true anymore if the manifolds are endowed with Riemannian metrics: a (curved) sphere is not locally isometric to a (flat) plane.Libermann began publishing research articles several years before completing the work for her thesis. For example she published, jointly with her supervisor Ehresmann,

*Sur les formes différentielles extérieures de degré 2*Ⓣ(1948),

*Sur le problème d'équivalence des formes différentielles extérieures quadratique*Ⓣ(1949), and

*Sur les structures presque hermitiennes isotropes*Ⓣ. She also published a number of single authored papers such as

*Problèmes d'équivalence relatifs à une structure presque complexe sur une variété à quatre dimensions*Ⓣ (1950),

*Sur la courbure et la torsion des variétés presque hermitiennes*Ⓣ (1951),

*Formes différentielles sur une variété symplectique*Ⓣ (1952), and

*Sur les variétés presque paracomplexes*Ⓣ (1952). In this early part of her career she spoke at a number of international conferences such as: 'Colloque de topologie et géométrie différentielle' Ⓣ held in Strasbourg in 1952; 'Géométrie différentielle' Ⓣ held at the Centre National de la Recherche Scientifique, Strasbourg in 1953; and 'Convegno Internazionale di Geometria Differenziale' held in Italy in 1953.

She was appointed Professor at the University of Rennes where she continued her research on differential geometry. During this time she was involved in a car accident as a passenger and never fully recovered from her injuries to the extent that for the rest of her life she had a stiff leg. In 1966 she was named Professor at the Faculty of Sciences at the University of Paris [12]:-

When the Faculty of Science of the University of Paris split after 1968, mostly along ideological lines, she chose Paris VII in accordance with her left-leaning convictions.Marc Chaperon writes [7]:-

She was quietly daring but also very faithful: to Élie Cartan of course, whose vision of the world was perpetuated by her seminar with Yvette Kosmann, during a period where his viewpoint was a bit out of fashion; and even more significantly to Ehresmann, gradually marginalised after having occupied a central focus in the mathematics of his time. At the Kosmann-Libermann seminar, the 'a fortiori' marginalised students of this great mind could express themselves. Paulette Libermann was in a good position to understand the meaning of both their work and exclusion. Perhaps due to the help that she had received from Élie Cartan as a beginner, she was always ready to welcome and encourage previously unknown young mathematicians, including Michèle Audin, Daniel Bennequin, Jean-Pierre Françoise and the author of this article [Marc Chaperon].Perhaps she is best known for her monograph

*Symplectic geometry and analytical mechanics*written jointly with Charles-Michel Marle. This was published in four volumes in French (1986, 1986, 1987, 1987) and an English text of 526 pages was published in 1987. The authors write in the Preface:-

During the last two centuries, analytical mechanics have occupied a prominent place among scientists' interests. The work in this field by such mathematicians as Euler, Lagrange, Laplace, Hamilton, Jacobi, Poisson, Liouville, Poincaré, Carathéodory, Birkhoff, Lie and Élie Cartan has played a major role in the development of several important branches of mathematics: differential geometry, the calculus of variations, the theory of Lie groups and Lie algebras, and the theory of ordinary and partial differential equations. During the last thirty years, the study of the geometric structures which form the basis of mechanics (symplectic, Poisson and contact structures) has enjoyed renewed vigour. The introduction of modern methods of differential geometry is one of the reasons for this renewal; it has permitted a formulation of global problems and furnished tools with which to solve them. Even though there are already a number of books that treat this subject, the authors believe that it is of value to provide readers with an approach to these methods and to permit them to familiarise themselves with certain recent developments which are not mentioned in the other textbooks in this field, and to acquire the information necessary in order to pursue current research. They have also expounded and employed the methods of exterior algebra which were introduced by E Cartan.Reviewing this book, I Vaisman writes [20]:-

The present work is an advanced textbook which gives a systematic exposition of the theory of symplectic, Poisson and contact manifolds, and their applications in Hamiltonian mechanics. ... The book, which was written by two well-known specialists with important contributions in the field (also reflected in this book) is very well written, updated, and it supplies a valuable contribution to the geometric-mechanical literature.N M J Woodhouse writes in the review [21]:-

The authors have clearly achieved their main aim: to give a systematic, accurate and largely self-contained account of the geometric foundations of classical mechanics for graduate students in mathematics. In doing so, they have written an excellent and lasting book.The book contains five chapters: Symplectic vector spaces and symplectic vector bundles; Semibasic and vertical differential forms in mechanics; Symplectic manifolds and Poisson manifolds; Action of a Lie group on a symplectic manifold; and Contact manifolds. For more information about this book, see THIS LINK.

Libermann was invited for extended research visits at many top universities. For example she spent time at St Hugh's College at Oxford working with Whitehead, at the University of California at Berkeley, at the Mathematical Sciences Research Institute in Berkeley, and at the National Institute of Pure and Applied Mathematics at Rio de Janeiro. She attended many international meetings where she was invited to lecture, only giving up travelling to conferences in 2004 when she was 85 years old. For example she gave a survey of various geometric concepts and results used in analytical mechanics in her lecture

*Liouville forms, parallelisms and Cartan connections*to the Jean Leray '99 Conference, and reviewed and summarised the theory of Cartan connections in her lecture

*Cartan connections and momentum maps*given at the 'Classical and Quantum Integrability' conference held in Warsaw in 2001. She continued to help to run the 'Geometry and Mechanics' seminar in Paris until the end of 2006. She broke her shoulder after a fall and then went to hospital for an operation in April 2007. She was taken to a retirement home at Montrouge, near Paris, following the operation but her health rapidly declined and she died in the home in July. Her final publication

*Charles Ehresmann's concepts in differential geometry*was published in 2007. Marc Chaperon writes [7]:-

Mademoiselle Libermann, as everyone called her, left us on 10 July 2007, after an exceptionally long and fruitful mathematical life. For example, till the end she actively participated in our seminar of Hamiltonian geometry, organised by her friend and collaborator Charles-Michel Marle; during the ensuing lunches, we benefited from her remarkable knowledge of mathematics and mathematicians but also from her stimulating vision of the world. I do not think she had changed substantially nor lost in vivacity since her admission in 1938 to the École Normale Supérieure de Jeunes Filles.Michèle Audin writes in [1]:-

Tiny, energetic, smiling, chatty, sometimes caustic, she was also a memory of the mathematical community. She liked to speak of those who helped her, either personally or professionally: Cartan's family, Jacqueline Ferrand, Ehresmann, anonymous others ... and she also liked to speak of those who did not help her. She participated, almost until the end, in conferences all around the world. The last time we met, in April, I was leaving for Vietnam. "I cannot go", she told me, "too tiring, I am getting old". She was 87.

### References (show)

- M Audin, Paulette Libermann,
*European Mathematical Society Newsletter***66**(2007), 19. - M Audin, Paulette Libermann. Héritage et descendance, Institut de Recherche Mathématique Avancée, University of Strasbourg (17 September 2010).

http://irma.math.unistra.fr/~maudin/Paulette.html - M Audin, Differential Geometry, Strasbourg, 1953,
*Notices Amer. Math. Soc.***55**(3) (2008), 366-370. - W M Boothby, Review: Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle,
*Bull. Amer. Math. Soc.***20**(1989), 89-94. - M Chaperon, En souvenir de Paulette Libermann,
*Gazette des Mathématiciens***122**(2009), 63-90. - M Chaperon, En souvenir de Paulette Libermann, II,
*Gazette des Mathématiciens***123**(2010), 49-66. - M Chaperon, Memories of Paulette Libermann,
*Eur. Math. Soc. Newsl. No.***77**(2010), 31-32. - M Chaperon, Souvenirs de Paulette Libermann. Un portrait mathématique de Paulette Libermann (1919-2007),
*Images des mathématiques, Le Centre national de la recherche scientifique*(2009).

http://images.math.cnrs.fr/Souvenirs-de-Paulette-Libermann.html?lang=fr - M de León and C Sardón, En memoria de Paulette Libermann,
*Matemáticas y sus fronteras*(16 July 2018).

https://www.madrimasd.org/blogs/matematicas/2018/07/16/145470 - A C Ehresmann, Sur Paulette Libermann (1919-2007),
*Cahiers de topologie et géométrie différentielle catégoriques***48**(4) (2007), 270-274. - Y Kosmann-Schwarzbach, Hommage à Paulette Libermann (1919-2007),
*SMF Gazette***114**(October 2007). - Y Kosmann-Schwarzbach, Women mathematicians in France in the mid-twentieth century,
*BSHM Bulletin: Journal of the British Society for the History of Mathematics***30**(3) (2015), 227-242. - E Kramer, Paulette Libermann, in
*The Nature and Growth of Modern Mathematics*(Hawthorn Books, Inc., New York, 1970), 711-712. - P Libermann and C-M Marle,
*Symplectic geometry and analytical mechanics*(Reidel Publishing Company, 1987). - C-M Marle, Hommage à Paulette Libermann (14 novembre 1919 - 10 juillet 2007),
*SMF Gazette***114**(October 2007). - C-M Marle, Homage to Paulette Libermann (Spanish),
*Gac. R. Soc. Mat. Esp.***10**(3) (2007), 643-645. - Paulette Libermann,
*Alchtron*(16 August 2018).

https://alchetron.com/Paulette-Libermann - T Ratiu, Review: Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle,
*American Scientist***78**(3) (1990), 282-283. - L Riddle, Paulette Libermann. November 14, 1919 - July 10, 2007,
*Biographies of Women Mathematicians, Agnes Scott College*(27 January 2020).

https://www.agnesscott.edu/lriddle/women/libermann.htm - I Vaisman, Review: Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle,
*Mathematical Reviews*MR0882548**(88c:58016)**. - N M J Woodhouse, Review: Symplectic geometry and analytical mechanics, by Paulette Libermann and Charles-Michel Marle,
*Bull. London Math. Soc.***20**(4) (1988), 377-379.

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Written by J J O'Connor and E F Robertson

Last Update December 2021

Last Update December 2021