Editing Projective differential geometry

{{short description|Geometry}}
In [[mathematics]], '''projective differential geometry''' is the study of [[differential geometry]], from the point of view of properties of mathematical objects such as [[function (mathematics)|function]]s, [[diffeomorphism]]s, and [[submanifold]]s, that are invariant under transformations of the [[projective group]]. This is a mixture of the approaches from [[Riemannian geometry]] of studying invariances, and of the [[Erlangen program]] of characterizing geometries according to their group symmetries. 

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 Čech]], amongst others), without a comprehensive theory of [[differential invariant]]s 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]], the simplest projective differential invariant.<ref name="Ovsienko2004">{{cite book|last=V. Ovsienko and S. Tabachnikov|title=Projective Differential Geometry Old and New From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups|year=2004|publisher=Cambridge University Press|isbn=9780521831864|page=vii (preface)|url=http://www.math.psu.edu/tabachni/Books/BookPro.pdf}}</ref>

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 [[curve]]s, 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.

==See also==
*[[Affine geometry of curves]]

==References==
{{reflist}}
* Ernest Julius Wilczynski ''[https://archive.org/details/projectivediffer00wilcuoft Projective differential geometry of curves and ruled surfaces]'' (Leipzig: B.G. Teubner,1906)

==Further reading==
*[https://web.archive.org/web/20100727082315/http://www.ima.umn.edu/imaging/SP7.17-8.4.06/activities/Eastwood-Michael/projective.pdf Notes on Projective Differential Geometry] by Michael Eastwood

[[Category:Differential geometry|*]]
[[Category:Projective geometry|*]]