By P. Ciarlet
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Extra resources for An Intro. to Differential Geometry With Applns to Elasticity
La topologie alggbrique des origines a Poincard° Paris: Presses Universitaires de France. ] Grundlagen ffir eine allgemeine Theorie der Functionen einer ver~nderlichen complexen Gr6sse. Inauguraldissertation GSttingen. Werke, 3-45. ] Uber die Hypothesen, welche der Geometrie zu Grunde liegen. Habilitationsvortrag GSttingen. GJttinger Abhandlungen 13 (1867). Werke, 272-287. ] Theorie der abelschen Funktionen. Journal fiir Mathematik 54. Werke 86-144. ". Werke, 391-404. ] Gesammelte mathematische Werke und wissenschaftlicher Nachlafl.
It is very easy to say how Klein saw non- Euclidean geometry fitting into this story. He did not. The language of nonEuclidean geometry is nowhere used, and it does not seem to have been considered. This is surprising, when one considers how natural it would have been to see the upper-half plane as representing non-Euclidean space. The situation with Poincar~ is very different. He himself described it once, when he talked to the Socigld de Psychologie in Paris in 19086. Not so many years ago I was fortunate enough to find three unpublished essays he wrote in 1880 which greatly elucidate the matter 7.
This was the purely mathematical aspect of his enterprise, which was kept in second place in his 1854-talk. The main goal of Riemann's inaugural lecture, on the other hand, was a reformulation of the conceptual foundations of physical geometry. This is made completely clear by the talk's architecture and the selection of main topics which culminate in a proposal of how the manifold concept could be used to analyze more deeply the properties of physical space. This goal also illuminates the reasons why Riemann choose manifolds of constant curvature as the only class of examples for a more detailed treatment in his second, differential geometric, part.