Georges de Rham | |
|---|---|
 Georges de Rham at the University of Geneva | |
| Born | (1903-09-10)10 September 1903 |
| Died | 9 October 1990(1990-10-09) (aged 87) Lausanne, Switzerland |
| Nationality | Swiss |
| Alma mater | University of Paris University of Lausanne |
| Known for | de Rham's theorem de Rham cohomology de Rham curve de Rham invariant Current Holonomy |
| Awards | Marcel Benoist Prize (1965) |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of Lausanne University of Geneva |
| Doctoral advisor | Henri Lebesgue |
Georges de Rham (French:[dəʁam]; 10 September 1903 – 9 October 1990) was aSwissmathematician, known for his contributions todifferential topology.
Georges de Rham was born on 10 September 1903 inRoche, a small village in the canton ofVaud inSwitzerland. He was the fifth born of the six children in the family of Léon de Rham, a constructions engineer.[1] Georges de Rham grew up in Roche but went to school in nearbyAigle, the main town of the district, travelling daily by train. By his own account, he was not an extraordinary student in school, where he mainly enjoyed painting and dreamed of becoming apainter.[2] In 1919 he moved with his family toLausanne in a rented apartment inBeaulieu Castle, where he would live for the rest of his life. Georges de Rham started theGymnasium in Lausanne with a focus on humanities, following his passion for literature and philosophy but learning little mathematics. On graduating from the Gymnasium in 1921 however, he decided not to continue with the Faculty of Letters in order to avoid Latin. He opted instead for the Faculty of Sciences of theUniversity of Lausanne. At the faculty he started out studying biology, physics and chemistry and no mathematics initially. While trying to learn some mathematics by himself as a tool for physics, his interest was raised and by the third year he abandoned biology to focus decisively on mathematics.[3]
At the University he was mainly influenced by two professors,Gustave Dumas andDmitry Mirimanoff, who guided him in studying the works ofÉmile Borel,René-Louis Baire,Henri Lebesgue, andJoseph Serret. After graduating in 1925, de Rham remained at the University of Lausanne as an assistant to Dumas. Starting work towards completing his doctorate, he read the works ofHenri Poincaré ontopology on the advice of Dumas. Although he found inspiration for a thesis subject in Poincaré, progress was slow as topology was a relatively new topic and access to the relevant literature was difficult in Lausanne.[2] With the recommendation of Dumas, de Rham contacted Lebesgue and went to Paris for a few months in 1926 and, again, for a few months in 1928. Both trips were financed by his own savings and he spent his time in Paris taking classes and studying at theUniversity of Paris and theCollège de France. Lebesgue provided de Rham with a lot of help in this period, both with his studies and supporting his first research publications. When he finished his thesis Lebesgue advised him to send it toÉlie Cartan and, in 1931, de Rham received his doctorate from the University of Paris before a commission led by Cartan and includingPaul Montel andGaston Julia as examiners.[1]
In 1932 de Rham returned to the University of Lausanne as an extraordinary professor. In 1936 he also became a professor at theUniversity of Geneva and continued to hold both positions in parallel until his retirement in 1971.[4]
De Rham was also one of the best mountaineers in Switzerland. As a member of the Independent High Mountain Group ofLausanne since 1944, he opened several difficult routes, some of them in theValais Alps (such as the south ridge of theStockhorn fromBaltschieder[5]) andVaud Alps (such asL'Argentine[6] and Pacheu). In 1944 he wrote a completeclimbing guidebook of theMiroir d'Argentine, where he climbed routes until 1980. According toJohn Milnor, in 1933 de Rham encountered on one of his hikesJames Alexander andHassler Whitney, who were climbing together near theWeisshorn inValais; this meeting was the beginning of a more than 40-year friendship between Whitney and de Rham.[7]
The theory ofdifferential forms has classical roots, with the relation between forms anddifferential topology initiated in the early 20th century byHenri Poincaré andÉlie Cartan, who observed thePoincaré lemma as well as the fact that not everyclosed differential form isexact. Cartan conjectured in 1928 that theBetti numbers of asmooth manifold could be encoded by differential forms. As a particular form of this, he conjectured that a closed form is exact if it integrates to zero over any submanifold without boundary, and that a submanifold without boundary is itself a boundary of another submanifold, if every closed form integrates to zero over it. De Rham, in his 1931 thesis, proved Cartan's conjecture by decomposing an arbitrary differential form into the sum of a closed form and some number ofelementary forms, which are differential forms associated to asmooth triangulation of the space.[8]
Following this work, de Rham made several attempts to unify forms and submanifolds into a single kind of mathematical object. He identified the ultimate notion of acurrent in the 1950s, generalizing (and inspired by)Laurent Schwartz's recent work ondistributions.[9] De Rham's work on these topics is now usually formulated in the language ofcohomology theory, although he did not do so himself.[8] In this form, his thesis work has become foundational to the field ofdifferential topology, while his theory of currents is basic togeometric measure theory and related fields.[10][11] His work is particularly important forHodge theory andsheaf theory.
In an additional part of his 1931 thesis, de Rham introduced higher-dimensional versions of the three-dimensionallens spaces and computed theirhomology, thereby establishing a necessary condition in order for two lens spaces to be homeomorphic.[8]
The structure of aRiemannian product automatically implies a product structure of theholonomy groups. In 1952 De Rham considered the converse, proving that, if there is a decomposition of thetangent bundle into vector subbundles which are invariant under the holonomy group, then the Riemannian structure must decompose as a product. This result, now known as thede Rham decomposition theorem, has become a fundamental textbook result inRiemannian geometry.[12][13]
