In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group.
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 invariants emerging.
[1] 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.