Projective differential geometry
- Projective differential geometry
In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties that are invariant under the projective group. This is a mixture of attitudes from Riemannian geometry, and the Erlangen program.
The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard ech, amongst others), without a comprehensive theory of differential invariants emerging. Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative.
Further work from the 1930s onwards was carried out by J. Kanitani, Shiing-Shen Chern, A. P. Norden, G. Bol, S. P. Finikov and G. F. Laptev. Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics and its applications, but the formulations given are still rooted in the language of the early twentieth century.
References
* Ernest Julius Wilczynski " [http://www.archive.org/details/projectivediffer00wilcuoft Projective differential geometry of curves and ruled surfaces] " (Leipzig: B.G. Teubner,1906)
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