Projective differential geometry

The branch of geometry in which one studies differential-geometric properties of curves and surfaces that are preserved under projective transformations. Such properties include, e.g., the concept of an asymptotic direction or, more generally, of conjugate directions, of an osculating quadric (in particular, a Lie quadric, a Darboux wreath of quadrics, etc.), of a projective normal, etc. The duality principle plays an important role in projective differential geometry. E.g., a surface in projective space can either be regarded as a two-parameter family of points or as the envelope of a two-parameter family of planes. The branches which were developed within projective differential geometry are: the (projective) theory of linear congruences (cf. Congruence in geometry), and problems on projective deformation and asymptotic transformation (in particular, the transformations of Båcklund, Bianchi, Eisenhart, Laplace, etc.).

The first study on projective differential geometry dates back to the end of the 19th century; the work of G. Darboux on surfaces and congruences was especially important. The first book in which classical projective differential geometry was systematically exposed is [1]. Subsequently appeared [2], [3], [4], in which projective differential geometry already appeared as an extensively developed geometric theory, related with other branches of geometry and with wide applications, e.g., in the theory of differential equations (in particular, non-linear, which made it possible to recently obtain "without quadratures" their solution by means of analogues of asymptotic transformations, see, e.g., Sine-Gordon equation).

G. Fubini and E. Čech gave an exposition of projective differential geometry in tensor form, using covariant differentiation, and obtained, basically, fundamental forms (cf. Fubini form, e.g.). In this way a solution was obtained for the problem of a projectively invariant framing of surfaces in three-dimensional space. S.P. Finikov and his school made major contributions to projective differential geometry, in particular regarding the theory of congruences and the theory of pairs of congruences and their transformations. The problem of a projectively invariant framing of a manifold in a higher-dimensional projective space was studied by G.F. Laptev and others.

An effective tool in the study of projective differential geometry of higher-dimensional spaces is the normalization method of A.P. Norden [6]. In this method one associates to each point $ x $ on an $ m $- dimensional surface $ X ^ {m} $ in a projective space $ P ^ {n} $ an $ ( n - m) $- dimensional plane incident with $ x $( a normal of the first kind), intersecting the tangent plane at $ x $ only, while in the $ m $- dimensional tangent plane one chooses an $ ( m - 1) $- dimensional plane (a normal of the second kind) not incident with $ x $. Moreover, on $ X ^ {m} $ there is an induced torsion-free affine connection, whose properties depend both on the structure of $ X ^ {m} $ as well as on the choice of normalization. In the case of a normalized hypersurface there arises a dual construction, leading to torsion-free intrinsic connections of the first and second kinds, dual with respect to the asymptotic tensor of the hypersurface. For special choices of normalization the differential geometry of spaces corresponding to subgroups of the projective group is included in the general scheme: affine, bi-axial, non-Euclidean, and Euclidean spaces.

Finally, in the work of E. Cartan a general theory of spaces with a projective connection was constructed. Laptev, using the method of exterior forms, has studied them as fibre spaces whose structure group is the group of projective transformations of projective space.

References

[1]	F. Wilczynski, "Geometria projectiva differenziale" Mém. Acad. Belgique , 3 : 2 (1911)
[2]	G. Fubini, E. Čech, "Geometria projettiva differenziale" , 1–2 , Zanichelli (1926–1927)
[3]	G. Bol, "Projective Differentialgeometrie" , 1–3 , Vandenhoeck & Ruprecht (1950–1967)
[4]	S.P. Finikov, "Projective differential geometry" , Moscow-Leningrad (1937) (In Russian)
[5]	S.P. Finikov, "Theorie der Kongruenzen" , Akademie Verlag (1959) (Translated from Russian)
[6]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[7]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian)

References

[a1]	E. Cartan, "Leçons sur la théorie des espaces à connexion projective" , Gauthier-Villars (1937)
[a2]	G. Fubini, E. Čech, "Introduction á la géométrie projective différentielle des surfaces" , Gauthier-Villars (1931)