Essay on the scientific investigations of German Fedorovich Laptev

G. F. Laptev (30.IV.1909–13.X.1972) was a well known Soviet geometer. This essay contains a short survey of his life, scientific and pedagogic activity and some more detailed exposition of his fundamental works in differential geometry. Chapter I deals with works of G. F. Laptev on manifolds of affine connection, intrinsically defined connections on surfaces imbedded in a space of affine connection (or merely in an affine space), deformation of surfaces preserving the complex of intrinsic connections on them (equivalent to the second order deformation in the sense of Fubini–Cartan in the case of surfaces in affine space). The main result of these works [3, 4, 6, 7] is the theorem of imbedding of a symmetric affine connection space $A_n$ in affine space $E_N$, $N=\frac{n^2+2n-4}2$.
G. F. Laptev is the author of a method used by many geometres, known, as the invariant method of differential geometric researches, based on the theory of external forms and Lie groups representations. This method is applied to the geometry of manifolds imbedded in Kleinian spaces with arbitrary generating element of in different generalized spaces. Chapter II contains an outline of the theory of geometric objects in the form given to it by G. F. Laptev, which is a fundation for his method. Varieties of fibred structure are dealt with in Chapter III. A variety of geometric elements is defined as a fibred space, the base of which is an arbitrary differentiable manifold and fibres are homogeneous spaces with geometric elements fixed in them. Connection is introduced as a map of a fibre onto infinitely neighboured fibre, satisfying some axioms. The sufficiency of the structural equations of E. Cartan for the existence of connection is proved (Theoreme of Cartan–Laptev). The theorem about the existence of regular prolongations of the fibred structure proved by Laptev (Ch. IIIl, § 1) allows construction of differential prolongations of arbitrary high order of the principal fibred spaces and of the associated fibred spaces. A global construction of structural forms and structural equations for principal fibre space and its differential prolongations is given. Chapter IV contains a short exposure of the method of Laptev in the theory of imbedded manifolds.
The scheme of constructing the differential geometry of any manifold, imbedded in a space of representation of a Lie group is in general terms as follows. The manifold is defined by a system of equations $a_{J^k}\theta^J=0$, where $\theta^J$ are the forms defining the representation space. The system of equations is prolonged successively by means of external differentiation and using E. Cartan's lemma. А sequence of representations is obtained (fundamental representations). Subordinated representations are constructed and studied. All the representations, constructed in this process, are fibre spaces and the initial manifold defines sections in them, which are regarded as fields of various geometric objects, invariantly related to the manifold. For a given manifold there exists generally such a field of fundamental object of order p which defines the manifold up to the transformations of the fundamental group. G. F. Laptev had applied his method to the geometry of $(n-1)$-surfaces and $m$-distributions in projective space $P_n$, affine $A_n$ and in spaces of projective and affine connection.
It is well known that such objects as Ehresrnann's jets, Bornpiani's calottas and Kawaguchi's higher order tensors are connected with differential prolongations of mappings and are intimately related to one another. The fundamental objects of mappings, introduced by G. F. Laptev give rise to all these notions in an invariant analytic form.