Projective deformation

An extension to projective geometry of the concept of deformation (superposition) in the metric theory of surfaces, given by G. Fubini in 1916 (a generalization of this concept to the geometry of any group of transformations was obtained by E. Cartan in 1920) using the concept of the so-called rolling of one surface over the other.

Let $ G $ be the group of transformations of a space $ E $. A surface $ S ^ \prime $ is superposed on a surface $ S $( or rolls over $ S $) in the geometry of the group $ G $ if a one-to-one correspondence is established between the points of $ S $ and $ S ^ \prime $ so that to each pair of corresponding points $ M \in S $ and $ M ^ \prime \in S ^ \prime $ a transformation $ y \in G $ can be assigned that takes $ S ^ \prime $ into the position $ S $. It is also required that

1) $ M ^ \prime $ be identified with $ M $;

2) every curve $ l ^ \prime \in S ^ \prime $ passing through $ M $ has at this point an $ n $- th order tangency with the corresponding curve $ l \in S $( that is, the distance between two points $ M ^ {\prime* } $ and $ M ^ {*} $ close to the common point $ M ^ \prime = M $ will be an infinitesimal of order $ n + 1 $ with respect to their distance from the common point). The correspondence between $ S $ and $ S ^ \prime $ characterized by the number $ n $ is called a superposition of order $ n $.

The contents of the notion of a distance here does not impose restrictions on the geometry of the group. However, here the order of tangency of curves is understood in a somewhat more narrow than usual sense of the word (the difference is that the correspondence between the points of the two curves is already established by the superposition, while usually it is established in defining the order of tangency).

Next, let $ G $ be the group of projective transformations and let $ S $ and $ S ^ \prime $ be projectively superposed. Then a projective deformation is a transformation of $ S $ preserving the projective line element

$$ ds = \frac{F _ 3}{F _ 2} , $$

where $ F _ {2} $ and $ F _ {3} $ are the Fubini forms (cf. Fubini form; in this case one has superposition of order two). And it turns out that besides ruled surfaces only the so-called $ R $- surfaces (see [1]) admit a non-trivial projective deformation.

Projective geometry occupies some middle position between metric geometry, where in general every surface can be deformed, and affine geometry, where the concept of deformation does not exist: Any two surfaces admit superposition of order one and no two different surfaces can have superposition of order two.

References

[1]	S.P. Finikov, "Projective-differential geometry" , Moscow-Leningrad (1937) (In Russian)
[2]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)

References

[a1]	S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)