ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53 Polyakova, Katerina, Kaliningrad State University, Russia Parallel displacements along manifold of osculating planes of a surface Surface X_{m} of a projective space P_{n} is considered as a manifold of triples (A, T_{m}, T_{s}), where A is a point of the surface, T_{m} is a tangent plane, T_{s} (s ≡ \frac{1}{2}m(m + 3) < n) is a osculating plane. Principle bundle G(X_{m}) associates with the surface and contains eight subbundles. In the bundle fundamental-group connection ? is given by G. F. Laptev’s way, eight subobject of which defines connections in the corresponding subbundles. Composition equipments of the surface X_{m} is made, that is next planes are joined to the point A: a) Norden’s normal of the second kind N_{m−1} : N_{m−1} ⊕ A = T_{m} ; b) P_{s−m−1} : T_{m} ⊕ P_{s−m−1} = T_{s};

c) P_{n−s−1} : T_{s} ⊕ P_{n−s−1} = P_{n}. Two types of scopes of the connection object ? and their coincidence conditions are found, the latter fix a hyperplane drawn on the all equipping planes. Parallel displacements of the equipping planes along lines on the surface are described. They characterise six subconnections defined by the subobjects of ? of both types. Three directions are considered. i. e. straight lines passing through the poin A and intersecting one from the equipping planes. A notion of the projective-covariant differentials introduced, it permits to interpret linear connections by parallel displacements of those directions.