MATHEMATICS

E. T. IVLEV

ON FRAMES OF SUBMANIFOLDS IN THE THEORY OF PAIRS OF COMPLEXES IN (P_3)

(Presented by Academician P. S. Aleksandrov on 17 III 1961)

In the present note the method of frames of submanifolds (\left({}^{1}\right)) is applied to the theory of pairs of complexes.

1. Let us consider, in three-dimensional projective space, a pair of complexes described by the corresponding rays (A_1A_2) and (A_3A_4). The points of intersection of these rays with the directrices of a linear congruence (L), belonging simultaneously to two linear complexes, each of which touches (\left({}^{4}\right)) one complex of the pair and passes through the corresponding ray of the other complex of this pair, are called (k)-flecnodal points. To obtain the canonical frame of an arbitrary submanifold, i.e., of a pair of nonholonomic congruences (\left({}^{2}\right)) belonging to the given pair of complexes, a semicanonical frame (\left({}^{1}\right)) of the pair of complexes is constructed. Placing the vertices of this frame (A_i) at the points harmonically dividing the (k)-flecnodal points of the corresponding rays of the pair of complexes, and carrying out invariant normalization, we obtain the derivational formulas:

[
dA_i=\omega_i^k A_k=\left(\alpha_i^k\omega_1^3+\beta_i^k\omega_2^4+\gamma_i^k\omega_3^1\right)A_k
\quad (i,k=1,2,3,4),
\tag{1}
]

where

[
\alpha_3^3=\beta_2^4=\gamma_3^1=1,\quad
\beta_1^3=\gamma_1^3=\alpha_2^4=\gamma_2^4=\alpha_3^1=\beta_3^1=0,
]

[
\alpha_1^1+\alpha_2^2+\alpha_3^3+\alpha_4^4
=
\beta_1^1+\beta_2^2+\beta_3^3+\beta_4^4
=
\gamma_1^1+\gamma_2^2+\gamma_3^3+\gamma_4^4=0,
]

[
\gamma_1^4=-\gamma_2^3=1,\quad
\alpha_1^4+\alpha_2^3=\beta_1^4+\beta_2^3,
]

[
\alpha_4^2(\gamma_3^2+\gamma_4^1)=(\gamma_2^4+1)(\alpha_3^2+\alpha_4^1),\quad
\beta_4^2(\gamma_3^2+\gamma_4^1)=(\gamma_4^2+1)(\beta_3^2+\beta_4^1).
]

In constructing the semicanonical frame of a pair of complexes, the following are excluded from consideration: a) pairs (T) of complexes (\left({}^{6,7}\right)); b) parabolic pairs of complexes, i.e., pairs of complexes with a parabolic linear congruence (L); c) pairs of special complexes. Exterior differentiation of the formulas leads to a system of 12 independent quadratic equations containing 29 unknown functions and defining the pair of complexes assigned to an arbitrary pair of nonholonomic congruences, with arbitrariness in 7 functions of 3 arguments.

2. Considering an arbitrary pair of nonholonomic congruences (\omega_i^k=0) ((i=1,\ldots,4;\ |i-k|=2)), belonging to the given pair of complexes, we obtain that the semicanonical frame of the pair of complexes is the canonical frame of this pair of nonholonomic congruences, since the point (A_i) on the ray of one complex describes a focal nonholonomic surface, while the point (A_j) ((|k-j|=1)) lies on the corresponding focal plane.

If one of the pairs of nonholonomic congruences (\omega_i^k=0) ((|i-k|=2)) is given, then the remaining ones are geometrically determined, and consequently the pairs of ruled surfaces belonging simultaneously to each two of these pairs of nonholonomic congruences are completely determined geometrically. Moreover, by assigning a coordinate pair of ruled surfaces, be-

belonging simultaneously to any two of the pairs of nonholonomic congruences (\omega_i^k=0) ((|i-k|=2)), are completely determined, as are the remaining pairs of nonholonomic congruences. In order to obtain the derivation formulas of the canonical frame of an arbitrary pair of nonholonomic congruences (\omega_1^3=0), it is sufficient in formulas (1) to put

[
\omega_1^3=0,\qquad
(\beta_i^k){\omega_1^3=0}=y_i^k,\qquad
(\gamma_i^k)=z_i^k .
\tag{2}
]

Since the frame is now geometrically determined, the coefficients (y_i^k) and (z_i^k) in the derivation formulas (1), under condition (2), are projective invariants of the pair of nonholonomic congruences. Among the functions (y_i^k) and (z_i^k) there are 21 independent ones. The coordinate pair of ruled surfaces (\omega_2^4=0) ((\omega_3^1=0)) of the given pair of nonholonomic congruences is characterized by the fact that the tangent plane at the point (A_2) ((A_3)) of the ruled surface described by the ray (A_1A_2) ((A_3A_4)) intersects the ray (A_3A_4) ((A_1A_2)) at the point (A_3) ((A_2)).

1. Various relations between the invariants (y_i^k) and (z_i^k) distinguish classes of pairs of nonholonomic congruences belonging to the given pair of complexes. A pair (h) of nonholonomic congruences has the natural equations (y_4^2=z_4^2=0) and is characterized by each of the following properties: 1) the vertex (A_4) is the focus of a ray of the nonholonomic congruence ((A_3A_4)), and the point (A_1) lies on the corresponding focal plane ((z_3^2y_4^1-y_3^2z_4^1\ne0)); 2) the points (A_1) and (A_4) are quasiflecnodal ({}^{(3)}) points of the corresponding rays of any pair of ruled surfaces belonging to (h). The coordinate pair of ruled surfaces (\omega_2^4=0) ((\omega_3^1=0)) of the pair (h), in addition to the properties noted above, is also characterized by the fact that the tangent plane at the (k)-flecnodal point (A_1\pm A_2) ((A_3\pm A_4)) of the ruled surface described by the ray (A_1A_2) ((A_3A_4)) intersects the ray (A_3A_4) ((A_1A_2)) at the (k)-flecnodal point (A_3\mp A_4) ((A_1\mp A_2)). The pair of nonholonomic congruences (y_3^2=y_4^1z_3^2-y_4^2=0) is characterized by the fact that the nonholonomic congruence ((A_3A_4)) is parabolic with focal nonholonomic surface described by the point (A_3). The pair of nonholonomic congruences (y_4^1=y_4^2z_3^2-y_3^2z_4^2=0) is characterized by the fact that the focal planes at the foci of a ray of the nonholonomic congruence ((A_3A_4)) intersect the ray (A_1A_2) at the points (A_1) and (A_2). The pair of nonholonomic congruences

[
(y_4^4-y_2^3)(z_4^2-z_3^2z_4^1)-y_4^2+y_3^2z_4^1+z_3^2y_4^1
= y_2^3y_4^1-y_4^1y_2^3(z_4^2-z_3^2z_4^1)=0
]

is characterized by the fact that the torses of the nonholonomic congruences ((A_1A_2)) and ((A_3A_4)) correspond. The pair of nonholonomic congruences (y_3^2=z_4^2y_4^1-y_4^2z_4^1=0) is characterized by the fact that the points (A_3) and (A_4) describe the focal nonholonomic surfaces of the nonholonomic congruence ((A_3A_4)). The pair of nonholonomic congruences

[
y_4^2-y_4^1+y_3^2=z_4^2-z_4^1+z_3^2+1=0
]

[
(y_4^2-y_3^2+y_4^1=z_4^2-z_3^2+z_4^1+1=0)
]

is geometrically characterized by the fact that the (k)-flecnodal point (A_3-A_4) ((A_3+A_4)) describes the focal nonholonomic surface of the parabolic nonholonomic congruence ((A_3A_4)).

1. The presence of submanifolds of a special type characterizes two most important pairs of complexes. The pair (R_1) of complexes, containing the submanifold (z_4^2=y_4^2=z_4^1+z_3^2+y_4^1+y_3^2=0), is geometrically characterized by each of the following properties: 1) the pair (h) of nonholonomic congruences forms a pair (T) (the pair (T) of nonholonomic congruences is defined in the same way as the pair (T) of holonomic congruences ({}^{(6)})); 2) in the pair (h) there are at least two stratifiable pairs of ruled surfaces; 3) the foci of the corresponding rays of the pair (h) are quasiflecnodal points of any pair of ruled surfaces of this pair.

A pair (R_2) of complexes containing the submanifold
(y_4^2=z_4^2=z_4^1+y_2^3=z_3^2+y_1^4=0) is characterized by each of the following properties: 1) a pair (h) of nonholonomic congruences of a stratification on both sides (a pair of nonholonomic congruences stratified on one or both sides is defined in the same way as a stratified pair of holonomic congruences ({}^{(6)})); 2) the compound ratio (\omega) (the compound ratio of the tangents to four curves on any focal nonholonomic surface of the pair (h) of nonholonomic congruences), corresponding to the coordinate pairs of ruled surfaces and torses of one nonholonomic congruence of the pair (h), is equal to the ratio corresponding to the coordinate pairs of ruled surfaces and torses of the other nonholonomic congruence of the pair (h); 3) the coordinate pairs of ruled surfaces of the pair (h) harmonically divide (i.e., (\omega=-1)) the stratified pairs of ruled surfaces of the same pair of nonholonomic congruences; 4) the stratified pairs of ruled surfaces of the pair (h) harmonically divide the pairs of ruled surfaces conjugate both in one and in the other nonholonomic congruences of the pair (h) (conjugate ruled surfaces of a nonholonomic congruence are defined in the same way as ruled surfaces of a holonomic congruence conjugate in the sense of Sannia ({}^{(5)})); 5) the stratified pairs of ruled surfaces of the pair (h) correspond to the asymptotic lines on all stratifying nonholonomic surfaces of both families. Pairs (R_2) of complexes form a subclass of the class of pairs (R_1) of complexes.

1. In the usual way we find that pairs (R_1) and (R_2) of complexes exist and are determined with arbitrariness in 4 and 3 functions of 3 arguments, respectively.

Tomsk State University
named after V. V. Kuibyshev

Received
14 III 1961

CITED LITERATURE

({}^{1}) R. N. Shcherbakov, Reports of the Scientific Conference on Theoretical and Applied Questions of Mathematics and Mechanics, Tomsk, 1960, p. 80.
({}^{2}) R. N. Shcherbakov, ibid., p. 82.
({}^{3}) E. T. Ivlev, ibid., p. 50.
({}^{4}) V. Hlavatý, Differentialní přímko va geometrie, Prague, 1941.
({}^{5}) R. N. Shcherbakov, Mathematical Collection, 46 (88), 2, 159 (1958).
({}^{6}) S. P. Finikov, Theory of Pairs of Congruences, Moscow, 1956.
({}^{7}) M. A. Akivis, Mathematical Collection, 27 (69), 351 (1950).