Differential Geometry, Strasbourg, 1953 Michèle Audin The picture on the following page, taken in 1953, difference between local and global geometry—the shows a group of mathematicians on the stairs of relation between global differential geometry and the historic Wilhelmian Building of the University topology is also one of the highlights in the intro- of Strasbourg. The picture appeared in the local duction of the conference proceedings. newspaper les Dernières Nouvelles d’Alsace to Bernard Malgrange remembers that the par- illustrate an article about a differential geom- ticipants had a good laugh at the declaration of etry conference organized by Charles Ehresmann Shiing-Shen Chern (1911–2004) to the journalist: (1905–1979), who was a professor at the Uni- “This conference is one of the main achievements versity of Strasbourg, and André Lichnerowicz of our time” (and this statement was quoted in (1915–1998), who was a professor at the Collège the article!). It seems indeed to have been a great de France. The photograph also appeared in the conference. The scientific program included André conference proceedings [6], which were published Weil (1906–1998) lecturing on nearby points on by the CNRS (Centre National de la Recherche Sci- smooth manifolds (sketching, he said, ideas of entifique). The two organizers stand in the center his master Nicolas Bourbaki—despite a conflict of the front row. André Lichnerowicz is the taller between master and pupil, as the latter explains of the two and easily recognizable by the pipe it in [12, p. 534]), Chern on infinite continuous from which he was inseparable (notice the way the pseudogroups, Jean-Louis Koszul on Lie groups stairs are used!). of transformations (the very first result stated in This photo inspired me to look back on some of his paper is the essential “slice theorem”), Nicolaas the important results achieved by the participants Kuiper (1920–1994) on locally affine surfaces, and in the years leading up to the conference and to Beno Eckmann on examples of compact complex consider the later impact of those results. The Kähler manifolds. René Thom (1923–2002) spoke photo also led me to muse on the lives and work on cobordant manifolds, defining the so-called of three mathematicians who were students in the Thom spaces M​O​(n​), M​S​O​(n​) and stating his Strasbourg School of Ehresmann, namely, Jacques results on the structure of the oriented and non- Feldbau, Georges Reeb, and Paulette Libermann. oriented cobordism rings Ω​∗​ and ​∗​, work that The newspaper article was published in the N would bring him the Fields Medal in 1958. Laurent Sunday May 31st/Monday June 1st edition, at the Schwartz (1915–2002), by then already a Fields end of a very busy week (the French president was Medalist, discussed the current associated with trying desperately to find a prime minister) and a meromorphic form on an analytic manifold. before another very busy week (the coronation of an English queen). We are lucky that there was The two organizers also gave talks. Ehresmann nevertheless some space left for a photo and an ar- explained the jet spaces he was inventing at that ticle about a mathematical conference. The article time, and two students of his, Paulette Libermann is unsigned. However, it is clear that the journalist and Georges Reeb, spoke as well. must have had some help in writing it, since he or This conference was not an accident that came out she wrote a few rather pertinent statements on the of nothing. There was tremendous activity around Ehresmann in Strasbourg, including an ongoing Michèle Audin is professor of mathematics at the Univer- seminar called “colloque de topologie de Strasbourg”. sité Louis Pasteur, Strasbourg, France. Her email address There remain written reports of the seminar talks, is [email protected]. given from 1951 to 1955, by speakers like Karl Stein, 366 NOTICES OF THE AMS VOLUME 55, NUMBER 3 Photograph: Dernières Nouvelles d’Alsace. Nouvelles Dernières Photograph: Differential geometry conference, University of Strasbourg, 1953. See photo key, page 369. Shoshichi Kobayashi, Kazumi Nomizu, Robert Her- one of the founders of the theory of fiber bundles. mann, Thom, Reinhold Remmert, André Haefliger, He is the one who first proved that a bundle over Reeb, Georges de Rham, Lawrence Markus, Liber- a simplex is trivializable and who used this to mann, and of course Ehresmann himself. classify bundles over spheres. This work appeared Charles Ehresmann had been a student of Élie in a Comptes-Rendus note [5] in 1939. Two years Cartan (1869–1951), who would have been seen by later, together with Ehresmann, he wrote two more the conference attendees as a kind of grandfather Compte-Rendus notes, in which the two authors of the field. As our journalist wrote, a huge number introduced the notion of an associated bundle and of new problems were investigated or sketched at proved results known today as the exact homotopy the conference, and many of them originated in the sequence of a fibration. work of Élie Cartan. Ehresmann, who defended his Two years later means 1941. And in the mean- thesis in 1934, was also a member of the Bourbaki time, there was the war (in which Feldbau served in group. He was very active and was one of the lectur- the French air force), the evacuation of Strasbourg ers in the so-called “Julia seminar” in 1935–37, the University to Clermont-Ferrand, the occupation of seminar that gave birth to the Bourbaki seminar France by Nazi Germany, the Anschluss of Alsace after the war. From 1939, Ehresmann taught at the (which had long been claimed as a part of Ger- University of Strasbourg, where he began to lead many) by the Third Reich, and the installation of an outstanding school of geometry and topology. the Pétain government, which very soon (October He is estimated to have directed more than sev- 1940) promulgated a set of antisemitic laws…So enty theses, including those of Haefliger and of Feldbau (who, as a Jew, was forbidden by the new Wu Wen Tsun. Let us now focus more closely on laws to teach), Ehresmann, and most of the other three of Ehresmann’s students in the Strasbourg professors and students of Strasbourg met again, school, Jacques Feldbau, Georges Reeb, and Pau- at the end of 1940, in Clermont-Ferrand, in the lette Libermann. center of France. And the first of the two joint notes by Ehresmann and Feldbau [4] appeared The One Who Is Missing in June 1941. The second one [3] was presented The very first student of Ehresmann was Jacques by Élie Cartan on October 27th but, at that time, Feldbau. Born in 1914 in Strasbourg, Feldbau was it was already impossible for the Académie des MARCH 2008 NOTICES OF THE AMS 367 Sciences to publish the name of a Jew…So the note appeared under the name of Ehresmann alone. It must have been very painful for Élie Cartan to remove Feldbau’s name from the paper before publication on December 1st and to reduce the credit given to Feldbau to a mention that the results were obtained by the au- thor “in collaboration with one of his students”. See [13, 1]. Feldbau could publish two Mathematisches the of courtesy Photo Oberwolfach. Forschungsinstitut short notes in the Bulletin de la Wilhelm Süss (left) and Georges Reeb in Société mathématique de France Oberwolfach, 1953. (which was edited by Henri Car- tan) under the pseudonym of was participating in the conference too. Reeb’s talk Jacques Laboureur (Feldbau in at the conference was about Finsler (and Cartan) German means agriculture, and Photograph courtesy of the Feldbau-Debré Family. Feldbau-Debré the of courtesy Photograph Laboureur is French for plow- spaces. This paper contains an intrinsic definition Jacques Feldbau. man). Feldbau was also one of of the Liouville form. the inventors of what is called Georges Reeb fit all the prejudices French people the Whitehead product between might have against Alsatians: he was massive and homotopy groups (because it was invented and slow, very slow. Slowness is not a quality French published by Whitehead at the same time). How- mathematicians usually praise very much. But Reeb ever, he could not publish his results because he did not care. He was even proud of it. He used to was caught in a Gestapo roundup in June 1943 and say that, when he was a professor at Grenoble sent to Auschwitz in October of that year. He sur- University (namely, at the time of our picture), the vived Auschwitz and even the deadly evacuation students attending his course suggested recording of the camp in January 1945 but eventually died of his lectures on tape, so that they could listen to exhaustion the following April, two weeks before them at double speed. When I arrived in Strasbourg the end of the war, in the Bavarian concentration in 1987, he used to come to the department and camp of Ganacker. See [2]. go from one office to another talking with people, This is why Jacques Feldbau—a handsome which had always been his way of working. Since young man with a very friendly and likeable per- he reckoned I was always in a hurry, he used to sonality, who practiced all kinds of sports (he was tell me the “story of the hurrying Savernois” (both a champion of the butterfly-stroke) and played the a paradox and a never-ending story). He also told piano, and who would have been a great topolo- many jokes about his complete baldness—a state gist—does not show up in the picture.