MATHEMATICS

V. S. MALAKHOVSKII

ON A CLASS OF CONGRUENCES OF CURVES OF THE SECOND ORDER WITH A DEGENERATING FOCAL SURFACE

(Presented by Academician P. S. Novikov on 14 IV 1962)

The paper considers congruences of curves of the second order \((^1)\) in three-dimensional projective space, possessing the following properties: 1) one focal surface of the congruence degenerates into a point; 2) there exist two nondevelopable focal surfaces \((B_{-1})\) and \((B_1)\); 3) the focal surfaces \((B_{-1})\) and \((B_1)\) are not the envelopes of the planes of the conics of the congruence; 4) the focal lines of each of these surfaces, together with the lines conjugate to them, form nets \(R\) \((^2)\), which are the second Laplace transform of one another. We shall agree to call these congruences \(C_1\)-congruences.

§ 1. Derivation formulas of the canonical frame. Focal surfaces and focal families of a \(C_1\)-congruence. Choose the vertices \(B_{-1}\) and \(B_1\) of the frame \(T(B_{-1}, B_0, B_1, A_2)\) at points describing the nets \(R\) on the focal surfaces \((B_{-1})\), \((B_1)\), the vertex \(B_0\) in their common first Laplace transform, and, finally, identify the vertex \(A_2\) with the fixed focal point of the congruence \(C_1\). With a certain normalization of the vertices, the derivation formulas of the canonical frame take the form:

\[ dB_{-1}=-b\omega^2P_0+a\omega^1P_1,\qquad dB_1=\omega^1P_0+\omega^2P_2,\qquad dA_2=0, \tag{1} \]

\[ \alpha B_0=-\omega^2B_{-1} +\frac12\left\{(1+ha^2)\omega^1+\left(a-2b-\frac1{ha}\right)\omega^2\right\}B_0 -ha^2b\omega^1B_1, \]

where

\[ P_0=B_{-1}+B_1-A_2,\qquad P_1=B_0+haB_{-1},\qquad P_2=B_0-aB_1. \tag{2} \]

Here \(\omega^1\) and \(\omega^2\) are Pfaffian forms satisfying the structural equations

\[ D\omega^1=(a-b)[\omega^1\omega^2],\qquad D\omega^2=\frac12(1-ha^2)[\omega^1\omega^2], \tag{3} \]

\(h\ne0\) is an arbitrary constant, and the invariants \(a\) and \(b\) satisfy the system of two ordinary differential equations

\[ 2\frac{d\ln a}{du}=1+ha^2-2hab,\qquad 2\frac{d\ln b}{du}=3(ha^2-1), \tag{4} \]

where \(du=\omega^1+\frac1{ha}\omega^2\) is a complete differential.

Consequently, the \(C_1\)-congruences exist and are determined up to a choice of three constants.

The conic of the \(C_1\)-congruence with respect to the canonical frame is determined by the equations

\[ x^1x^{-1}+x^{-1}x^2+x^2x^1=0,\qquad x^0=0. \tag{5} \]

The foci, distinct from \(B_{-1}, B_1\), and the focal families that are not the lines \(\omega^1\omega^2=0\), are found from equations (5) and the equations

\[ a\omega^1x^{-1}+\omega^2x^1=0,\qquad \omega^2(a-2b)x^{-1}+(2-ha^2)\omega^1x^1=0. \tag{6} \]

Theorem 1. The congruence \(C_1\) has four nondegenerate focal surfaces and four focal families.

§ 2. Geometric properties of the congruence \(C_1\).

Denote by \(B_{-n}, B_n\) the \(n\)-th Laplace transform of the surface \(B_0\), respectively in the direction of the lines \(\omega^1=0,\ \omega^2=0\). For any natural number \(n\), the following formulas hold:

\[ \begin{aligned} B_2&=B_{-1}-A_2, && B_{-2}=A_2-B_1,\\ B_{3n}&=B_0+ahP_{n-1}(h)A_2, && B_{-3n}=B_0-ah^{1-n}P_{n-1}(h)A_2,\\ B_{1+3n}&=B_1+hP_{n-1}(h)A_2, && B_{-(1+3n)}=B_{-1}+h^{-n}P_{n-1}(h)A_2,\\ B_{2+3n}&=B_{-1}-P_n(h)A_2, && B_{-(2+3n)}=B_1-h^{-n}P_n(h)A_2, \end{aligned} \tag{7} \]

where \(P_n(h)=h^n+h^{n-1}+\cdots+h+1\).

From formulas (7) the following theorems follow:

Theorem 2. If the natural numbers \(n\) and \(n+1\) are not congruent to the number \(m\) modulo \(3\) \((m=0,1,2)\), then the line \(B_nB_{n+1}\) passes through the point \(P_m\).

Theorem 3. If \(k\) and \(k-1\) are negative integers such that the nonnegative numbers \(3n+k\) and \(3n+k-1\) (\(n\) natural) are not congruent to the number \(m\) modulo \(3\) \((m=0,1,2)\), then the line \(B_kB_{k-1}\) passes through the point \(P_m\).

Theorem 4. If, for an arbitrary integer \(k\), we have

\[ k\equiv p \pmod 3,\qquad p=0,\ \pm1,\ \pm2, \]

then the Laplace transform \(B_k\) lies on the line \(AB_p\).

Theorem 5. The cross ratios \((B_0A;\,B_{3n}B_{-3n})\), \((B_1A;\,B_{1+3n}B_{1-3n})\), \((B_{-1}A;\,B_{-1+3n}B_{-1-3n})\) are equal, with the opposite sign, to the \(n\)-th power of the invariant \(h\).

Analyzing the properties of the three-vertex figure \(P_0,P_1,P_2\), we see that its sides touch the lines \(\omega^1=0\) and \(\omega^2=0\) at the corresponding vertices. The plane \((P_0,P_1,P_2)\) is stationary. The lines \(\omega^1=0\), \(\omega^2=0\), \(\omega^1+\dfrac{1}{ha}\omega^2=0\) form on it three nets described by the vertices \(P_0,P_1\), and \(P_2\).

§ 3. Some subclasses of congruences \(C_1\).

1) Congruences \(h=-1\). They are determined with arbitrary two constants and are characterized by the fact that \(B_{-3}=B_3\), i.e., the Laplace sequence \(\{B_{-p},B_q\}\) is closed with period 6.

2) Congruences \(h=1\). They are also determined with arbitrary two constants and are characterized by the fact that all quadruples of points indicated in Theorem 5 are harmonic. The asymptotic lines of one family of all surfaces \(R\) of the congruence \(h=1\) are lines with constant invariants and are mapped by the rays of rectilinear congruences onto the lines of the stationary plane \((P_0,P_1,P_2)\) passing through the fixed focal point of the congruence \(C_1\).

3) Congruences \(a=b=h^{-1/2}\). They are determined with arbitrary one constant and are the only congruences \(C_1\) with constant invariants.

Tomsk State University
named after V. V. Kuibyshev

Received
9 IV 1962

CITED LITERATURE

1. N. G. Tuganov, DAN, 100, No. 1 (1955).
2. S. P. Finikov, Theory of Congruences, Moscow—Leningrad, 1950.