In
mathematics, projective differential geometry is the study of
differential geometry, from the point of view of properties of mathematical objects such as
functions,
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given tw ...
s, and
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s, that are invariant under transformations of the
projective group. This is a mixture of the approaches from
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...
of studying invariances, and of the
Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
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 invariants emerging.