MATHEMATICS

K. I. DUNICHEV

FIBRATION OF TWO-PARAMETER FAMILIES OF STRAIGHT LINES BY LINES IN A MULTIDIMENSIONAL PROJECTIVE SPACE

(Presented by Academician P. S. Novikov, 24 X 1962)

The fibration of manifolds by families of lines was proposed in 1945 by S. P. Finikov and applied to the construction of a fibered pair of line-like surfaces \((^{1})\) and complexes of straight lines \((^{2})\). In our article \((^{3})\) this notion was generalized to pairs of manifolds with generating elements of different dimension in four-dimensional projective space. In the present work we carry out the fibration by families of lines of a pair of two-parameter families of straight lines in \(n\)-dimensional projective space \(P_n\) \((n>3)\).

1. In the space \(P_n\) consider a pair of families \(\infty^2\) of straight lines \(\{l\}\) and \(\{l'\}\), whose elements are in one-to-one correspondence. With each straight line \(l\) of one family we associate a pencil of plane elements (two-dimensional planes), the centers of which lie on \(l\), while the axis of the pencil is \(l'\). We shall call the pair \(\{l,l'\}\) one-sidedly fibered in the direction from \(\{l\}\) to \(\{l'\}\) if the plane elements are tangent planes to the lines described by the centers of these elements as \(l\) moves within the subfamily \(\{l\}\). The lines described by the centers of the plane elements will be called the fibering lines. If the pair \(\{l,l'\}\) is fibered both in the direction from \(\{l\}\) to \(\{l'\}\) and in the direction from \(\{l'\}\) to \(\{l\}\), then such a pair will be called two-sidedly fibered, or simply a fibered pair.

We shall assume that the corresponding elements of the families \(\{l\}\), \(\{l'\}\) do not intersect and that the family \(\{l\}\) is focal, possessing two distinct focal surfaces.

1. Assuming that only one focal plane of the ray \(l\) intersects the ray \(l'\), place the vertices \(A_1\) and \(A_2\) of a moving frame at the foci of the ray \(l\), the vertex \(A_3\) at the point of intersection of one of the focal planes of the ray \(l\) with the ray \(l'\), the vertex \(A_4\) arbitrarily on \(l'\), and the vertex \(A_5\) in the second focal plane of the ray \(l\). We do not fix the positions of the remaining vertices \(A_\alpha\) \((\alpha=6,7,\ldots,n+1)\) of the frame. The infinitesimal displacements of the vertices of the frame are determined by the equations \(dA_i=\omega_i^j A_j\), where the forms \(\omega_i^j\) satisfy the structure equations of projective space \(D\omega_i^j=[\omega_i^k\omega_k^j]\) \((i,j,k=1,2,\ldots,n+1)\). With the indicated choice of the vertices of the frame, the family \(\{l\}\) is determined by the equations

\[ \omega_1^4=\omega_1^5=\omega_1^\alpha=\omega_2^3=\omega_2^4=\omega_2^\alpha=0,\qquad [\omega_1^3\omega_2^5]\ne 0 \tag{1} \]

and depends on an arbitrary two functions of two arguments. The developable surfaces of the family are determined by the equations \(\omega_1^3=0\) and \(\omega_2^5=0\). The tangent plane of the focal surface \((A_1)\) intersects the ray \(l'\) at the point \(A_3\).

The point \(M=A_1+\rho A_2\) describes a fibering line \((M)\) if along this line

\[ (dM,\ M,\ A_3,\ A_4)=0,\qquad (d^2M,\ M,\ A_3,\ A_4)=0. \]

The first of these conditions gives the equations of the ruling lines \((M)\):

\[ \omega^5_2=0,\qquad d\rho+\rho(\omega^2_2-\omega^1_1)+\omega^2_1-\rho^2\omega^1_2=0, \]

the second gives the conditions of a one-sided fibration:

\[ \omega^1_3\equiv 0,\qquad \omega^2_3\equiv 0,\qquad \omega^5_3\equiv 0,\qquad \omega^\alpha_3\equiv 0 \pmod{\omega^5_2}. \tag{2} \]

The point \(M'=A_3+\rho' A_4\) describes a ruling line \((M')\), if along this line

\[ (dM',\,M',\,A_1,\,A_2)=0,\qquad (d^2M',\,M',\,A_1,\,A_2)=0. \]

Whence, in addition to equations (2), we obtain

\[ \omega^1_4=0,\qquad \omega^5_4\equiv 0,\qquad \omega^\alpha_4=0 \pmod{\omega^5_2}, \tag{3} \]

and the ruling lines \((M')\) are determined by the equations

\[ \omega^5_2=0,\qquad d\rho'+\rho'(\omega^4_4-\omega^3_3)+\omega^4_3-(\rho')^2\omega^3_4=0. \]

Equations (1), (2), (3) and their exterior differentials constitute the system of equations of the problem.

Theorem 1. A fiberable pair \(\{l,l'\}\) exists with an arbitrary number \(\varepsilon\): two functions of two arguments.

By virtue of the one-sided fibration, the family \(\{l'\}\) becomes focal: the unique (without additional restrictions) focal surface is described by the point \(A_3\) of intersection of one of the focal planes of the ray \(l\) with the ray \(l'\); the developable surface is defined by the equation \(\omega^5_2=0\). The families of ruling lines \((M)\) and \((M')\) correspond and lie in the developable surfaces of the families \(\{l\}\), \(\{l'\}\). The lines \((M)\), \((M')\) are three-dimensional, belong to the space stretched over the lines \(l,l'\). The tangents to the ruling lines \((M)\) at the points of the ray \(l\) pass through the focus \(A_3\) of the ray \(l'\), while the tangents to the ruling lines \((M')\) at the points of the ray \(l'\) pass through the focus \(A_2\) of the ray \(l\).

3. Theorem 2. If both focal planes of the ray \(l\) intersect the ray \(l'\), then under a one-sided fibration in the direction from \(\{l\}\) to \(\{l'\}\), either the pair \(\{l,l'\}\) is immersed in a three-dimensional space, or the ruling lines \((M)\) become straight lines.

4. Theorem 3. If both focal planes of the ray \(l\) do not intersect the ray \(l'\), then the ruling lines \((M)\) do not exist.

5. Let us summarize the above in the form of a theorem:

Theorem 4. Let \(\{l\}\) and \(\{l'\}\) be two-parameter families of lines in \(P_n\), and let \(\{l\}\) possess two distinct focal surfaces. A fibration of the pair \(\{l,l'\}\) by families of lines in \(P_n\) \((n>3)\) is possible if and only if only one focal plane of the ray \(l\) intersects the ray \(l'\).

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
19 X 1962

CITED LITERATURE

1. S. P. Finikov, Izv. Akad. Nauk SSSR, Ser. Mat., 9, 79 (1945).
2. S. P. Finikov, UMN, 9, 1 (59), 125 (1954).
3. K. I. Dunichev, Izv. vyssh. uchebn. zaved., Ser. Mat., No. 1 (2), 43 (1958).