MATHEMATICS

G. B. KHASIN

SUCCESSIVE LAPLACE TRANSFORMATIONS WITH FIBERING CONGRUENCES OF AXES

(Presented by Academician P. S. Aleksandrov, March 28, 1962)

1. Statement of the problem. Let, in three-dimensional projective space, \((M_0)\) be a net of conjugate lines, and \((M_1)\) its first Laplace transform. The line joining the first Laplace transforms of the net is called its second axis, and the line along which the tangent planes to the first Laplace transforms of the net intersect is called the first axis of this net \((^1)\). We shall find a net \((M_0)\) for which the congruence of first axes fibers with the congruence of first axes of the net \((M_1)\), and, at the same time, the congruences of second axes of the nets \((M_0)\) and \((M_1)\) fiber.

2. Equations of the problem. Let \((M_0)\) be the required conjugate net, \((M_1)\) its first Laplace transform, and \((M_{-1})\) the other Laplace transform of the net \((M_0)\). Denote the Laplace transform of the net \((M_1)\) by \((N_2)\). Since the first axis of the net \((M_0)\) lies in the plane tangent to the net \((M_1)\), this axis intersects the line \(M_1N_2\). Denote the point of intersection by \(M_3\). Similarly, the first axis of the net \((M_1)\) intersects \(M_0M_{-1}\); let \(M_2\) be the point of intersection of \(M_0M_{-1}\) with this axis.

The equations of infinitesimal displacements of the vertices of the moving frame are written in the form \(dM_i=\omega_i^j M_j\) \((i,j=0,1,2,3)\), where the differential forms \(\omega_i^j\) satisfy the structure equations \((^2)\)
\(D\omega_i^j=[\omega_i^k\omega_k^j]\).

If the net \((M_0)\) is referred to the frame \([M_0M_1M_2M_3]\), then the infinitesimal forms satisfy the system

\[ \begin{aligned} \omega^3&=0, & \omega_2^1&=0,\\ \omega_1^3&=a\omega^1, & \omega_3^2&=\sigma\omega^1,\\ \omega_2^3&=c\omega^2, & \omega_3^0&=\nu\omega_1^0,\\ \omega_1^2&=0, & \omega_1^0&=\beta\omega^2. \end{aligned} \tag{1} \]

Here \(M_{-1}=kM_0-M_2;\quad N_2=\nu M_1-M_3\).

We require that the congruence of axes \([M_0M_3]\) fiber with the congruence of axes \([M_1M_2]\) \((^3)\).

The equations for the fibering of the axes \([M_0M_3]\) and \([M_1M_2]\) have the form

\[ [\omega_3^1\omega_1^0]+[\omega_3^2\omega_2^0]=0, \]

\[ [\omega_2^0\omega^2]-[\omega_2^3\omega_3^2]+[\omega_1^0\omega^1]-[\omega_1^3\omega_3^1]=0; \tag{2} \]

\[ [\omega_2^0\omega^1]+[\omega_2^3\omega_3^1]=0, \]

\[ -[\omega^1\omega_1^0]+[\omega_2^0\omega_2^0]-[\omega_3^1\omega_1^3]+[\omega_3^2\omega_2^3]=0. \tag{3} \]

Systems (2) and (3) give

\[ \begin{aligned} [\omega_3^1\omega_1^0]+[\omega_3^2\omega_2^0]&=0, &&(\text{a})\\ [\omega_2^0\omega^1]+[\omega_2^3\omega_3^1]&=0, &&(\text{б})\\ [\omega_2^0\omega_2^0]+[\omega_1^3\omega_3^1]&=0, &&(\text{в})\\ [\omega^1\omega_1^0]+[\omega_2^3\omega_3^2]&=0. &&(\text{г}) \end{aligned} \tag{4} \]

Together with system (1), equation (4г) gives \(\alpha=c\sigma\), and equations (4а) and (4б) are equivalent. Consequently, (4), by virtue of (1), gives

\[ \beta=c\sigma; \qquad a[\omega_3^1\omega^1]+[\omega_2^0\omega^2]=0;\qquad [\omega_2^0\omega^1]-c[\omega_3^1\omega^2]=0. \tag{5} \]

The conditions for the stratification of the congruences of the second axes, taking into account (1) and (5), lead to the systems:

\[ \{[\Delta k\omega^1]=0,\qquad [\Delta \nu\omega_1^3]+[\Delta k;\omega^2]=0\}, \tag{6} \]

this is the condition for the stratification of \([M_0N_2]\) onto \([M_1M_{-1}]\);

\[ \{[\Delta \nu\omega^2]=0,\qquad [\Delta \nu\omega_1^3]+[\Delta k\omega^2]=0\}, \tag{6'} \]

this is the condition for the stratification of \([M_1M_{-1}]\) onto \([M_0N_2]\); here

\[ \Delta k=dk+k(\omega_0^0-\omega_2^2)+k^2\omega^2;\qquad \Delta \nu=d\nu+\nu(\omega_1^1-\omega_3^3)+\nu^2\omega_1^3. \]

If one differentiates the equations \(\omega_3^1=k\omega^1\) and \(\omega_2^0=\nu\omega_1^0\), then from system (1) we shall have \([\Delta k\omega^1]=0;\ [\Delta \nu\omega^2]=0\). Thus, the first equations of systems (6) and \((6')\) are satisfied by virtue of system (1), and the second equations of these systems are identical. Thus we obtain:

Theorem. If the congruences of the first axes of the net \((M_0)\) and the net \((M_1)\) stratify, and the congruences of the second axes stratify only in one direction, then the congruences of the second axes of the nets \((M_0)\) and \((M_1)\) stratify in the other direction as well.

Combining systems (1), (5), (6), and \((6')\), we obtain all the equations of the problem:

\[ \begin{array}{lll} \beta=c\sigma, & \text{(а)} & \omega_2^1=k\omega^1, \quad \text{(е)} \qquad [\omega_2^0\omega^1]-c[\omega_3^1\omega^2]=0, \quad \text{(л)}\\ \omega^3=0, & \text{(б)} & \omega_3^2=\sigma\omega^1, \quad \text{(ж)} \qquad a[\omega_3^1\omega^1]+[\omega_2^0\omega^2]=0, \quad \text{(м)}\\ \omega_1^3=a\omega^1, & \text{(в)} & \omega_1^0=\beta\omega^2, \quad \text{(з)} \qquad [\Delta k\omega^1]=0, \quad \text{(н)}\\ \omega_2^3=c\omega^2, & \text{(г)} & \omega_3^0=\nu\omega_1^0, \quad \text{(и)} \qquad [\Delta \nu\omega^2]=0, \quad \text{(о)}\\ \omega_1^2=0, & \text{(д)} & d\beta=c\,d\sigma+\sigma\,dc, \quad \text{(к)} \qquad [\Delta k\omega^2]+[\Delta \nu\omega_1^3]=0. \quad \text{(п)} \end{array} \tag{7} \]

3. Existence of a solution.

Differentiating equations (7б)—(7и), we have

\[ \begin{array}{ll} [\Delta \ln a\omega^1]=0, & \Delta\ln a=d\ln a+\omega_0^0-2\omega_1^1+\omega_3^3+k\omega^2, \quad \text{(а)}\\ [\Delta \ln c\omega^2]=0, & \Delta\ln c=d\ln c+\omega_0^0-2\omega_2^2+\omega_3^3, \quad \text{(б)}\\ [\Delta \ln \sigma\omega^1]=0, & \Delta\ln\sigma=d\ln\sigma+\omega_0^0-\omega_1^1+\omega_2^2-\omega_3^3, \quad \text{(в)}\\ [\Delta \ln \beta\omega^2]=0, & \Delta\ln\beta=d\ln\beta+2\omega_0^0-\omega_1^1-\omega_2^2+\nu\omega_1^3. \quad \text{(г)} \end{array} \tag{8} \]

Let us note that \(a\ne0,\ c\ne0,\ \beta\ne0,\ \sigma\ne0\), since otherwise the net \((M_1)\) would degenerate.

Substituting \(\Delta\ln c\), \(\Delta\ln\beta\), \(\Delta\ln\sigma\) into (7к), we obtain \(\Delta\ln\beta=\Delta\ln\sigma+\Delta\ln c-k\omega^2-\nu\omega_1^3\), which by virtue of (8г) gives \([\Delta\ln\sigma-a\nu\omega^1;\omega^2]=0\). Together with (8в) this gives \(\Delta\ln\sigma=a\nu\omega^1\), or \(d\ln\sigma=\nu\omega_1^3-k\omega^2-\omega_0^0+\omega_1^1-\omega_2^2+\omega_3^3\). We have obtained a Pfaff equation, which must be adjoined to system (7) and differentiated exteriorly. Differentiating, we obtain \([\Delta\nu\omega_1^3]-[\Delta k\omega^2]=0\). Together with equation (7п) this gives \(\Delta\nu=0,\ \Delta k=0\). Differentiating these equations exteriorly, we obtain

\[ k\{[\omega_1^0\omega^1]+[\omega_2^0\omega^2]\}=0,\qquad \nu\{[\omega_1^0\omega^1]+[\omega_2^0\omega^2]\}=0. \]

These equations allow the whole investigation to be divided into two cases:

\[ \text{I. }\ [\omega_1^0\omega^1]+[\omega_2^0\omega^2]=0;\qquad \text{II. } k=\nu=0. \]

I. Let us consider the case \([\omega^0_1\omega^1]+[\omega^0_2\omega^2]=0\). The system of equations of the problem has the form

\[ \begin{gathered} \omega^3=0,\qquad \omega^2_3=\sigma\omega^1,\qquad [\omega^0_2\omega^1]-c[\omega^1_2\omega^2]=0,\\ \omega^3_1=a\omega^1,\qquad \omega^1_2=k\omega^1,\qquad a[\omega^1_3\omega^1]+[\omega^0_2\omega^2]=0,\\ \omega^3_2=c\omega^2,\qquad \omega^0_3=\nu\omega^0,\qquad [\omega^0_1\omega^1]+[\omega^0_2\omega^2]=0,\\ \omega^2_1=0,\qquad \omega^0_1=c\sigma\omega^1,\qquad dk=k(\omega^2_2-\omega^0_0)-k^2\omega^2,\\ d\nu=\nu(\omega^3_3-\omega^1_1)-\nu^2\omega^3_1. \end{gathered} \tag{7′} \]

The system of quadratic equations has the form

\[ \begin{gathered} [\Delta\ln a\,\omega^1]=0,\qquad [\omega^0_2\omega^1]-c[\omega^1_3\omega^2]=0,\\ [\Delta\ln c\,\omega^2]=0,\qquad a[\omega^1_3\omega^1]+[\omega^0_2\omega^2]=0,\\ [\omega^0_2\omega^2]+[\omega^0_1\omega^1]=0. \end{gathered} \tag{9} \]

Expanding equations (9) by Cartan’s lemma, we obtain

\[ \begin{gathered} \Delta\ln a=A_1\omega^1\quad \text{or}\quad d\ln a=A_1\omega^1-\omega^0_0+2\omega^1_1-\omega^3_3-k\omega^2, \quad (a)\\ \Delta\ln c=A_2\omega^2\quad \text{or}\quad d\ln c=A_2\omega^2-\omega^0_0+2\omega^2_2-\omega^3_3, \quad (\text{б})\\ \omega^1_3=-aA_3\omega^1+\beta\omega^2, \quad (\text{в})\\ \omega^0_2=\beta\omega^1+cA_3\omega^2. \quad (\text{г}) \end{gathered} \tag{10} \]

Here \(N=3\), and since \(q=4\), \(Q>N\), and the system (7′) is not in involution. We must adjoin equations (10) to the system (7) and consider the prolonged system. The differential consequences of the system (7) are satisfied by virtue of equations (10). Therefore only equations (10) should be differentiated exteriorly. Differentiating (10a) and (10б), we shall have

\[ [\Delta A_1\omega^1]=0,\qquad \text{where } \Delta A_1=dA_1+A_1(\omega^0_0-\omega^1_1+k\omega^2); \]

\[ [\Delta A_2\omega^2]=0,\qquad \text{where } \Delta A_2=dA_2+A_2(\omega^0_0-\omega^2_2). \]

From equations (10в) and (10г) we have

\[ dA_3+A_3(\omega^0_0-\omega^3_3)+kA_3\omega^2-\frac{\beta}{a}A_1\omega^2+2\beta\nu\omega^2-k\sigma\omega^1-\sigma A_2\omega^1=0. \]

Adjoining this equation to the system (10), we obtain

\[ -\frac{\beta}{a}[\Delta A_1\omega^1]+\frac{\beta A_1^2}{a}[\omega^1\omega^2] -\sigma[\Delta A_2\omega^1]+\sigma kA_2[\omega^1\omega^2]-\beta\nu A_1[\omega^1\omega^2]=0. \]

Thus the prolonged system has the consequences:

\[ [\Delta A_1\omega^1]=0,\qquad [\Delta A_2\omega^2]=0, \]

\[ \frac{\beta}{a}[\Delta A_1\omega^2]+\sigma[\Delta A_2\omega^1] +\left(\beta\nu A_1-k\sigma A_2-\frac{\beta A_1^2}{a}\right)[\omega^1\omega^2]=0. \tag{11} \]

Expanding by Cartan’s lemma, we obtain

\[ \begin{gathered} \Delta A_1=B_1\omega^1\quad \text{or}\quad dA_1=B_1\omega^1-A_1(\omega^0_0-\omega^1_1)-A_1k\omega^2, \quad (a)\\ \Delta A_2=B_2\omega^2\quad \text{or}\quad dA_2=B_2\omega^2-A_2(\omega^0_0-\omega^2_2), \quad (\text{б})\\ cB_1=aB_2+cA_1^2+akA_2-ac\nu A_1. \quad (\text{в}) \end{gathered} \tag{12} \]

Here \(N=1\), and since \(q=2\), \(Q>N\). Differentiating (12в), we obtain

\[ \begin{aligned} cdB_1={}&adB_2+aB_2(A_1\omega^1-A_2\omega^2)+2aB_2(\omega^1_1-\omega^2_2)+2cA_1B_1\omega^1\\ &-2cA_1^2(\omega^0_0-\omega^1_1)-2cA_1^2k\omega^2-2akA_2(\omega^0_0-\omega^1_1)+akA_1A_2\omega^1\\ &-akA_2^2\omega^2-2ak^2A_2\omega^2-ac\nu B_1\omega^1+2ac\nu kA_1\omega^2\\ &+a^2c\nu^2A_1\omega^1-ac\nu A_1^2\omega^1+2ac\nu A_1(\omega^0_0-\omega^1_1). \end{aligned} \tag{13} \]

Equations (12a) and (12b) give

\[ [\Delta B_1 \omega^1]=0,\qquad \text{where } \Delta B_1=dB_1+2B_1(\omega^0_0-\omega^1_1+k\omega^2), \tag{13'} \]

\[ [\Delta B_2 \omega^2]=0,\qquad \text{where } \Delta B_2=dB_2+2B_2(\omega^0_0-\omega^2_2). \]

From equations \((13')\) it follows that \(\Delta B_1=L_1\omega^1;\ \Delta B_2=L_2\omega^2\). Substituting this into (13), we obtain

\[ L_1=3A_1B_1-A_1^3-avB_1+a^2v^2A_1,\qquad L_2=A_2B_2-2kB_2+kA_2^2. \]

Thus, system (13) gives

\[ \begin{aligned} dB_1&=(3A_1B_1-A_1^3-avB_1+a^2v^2A_1)\omega^1 -2B_1(\omega^0_0-\omega^1_1+k\omega^2),\\ dB_2&=(A_2B_2-2kB_2+kA_2^2)\omega^2 -2B_2(\omega^0_0-\omega^2_2). \end{aligned} \tag{14} \]

System (14) is completely integrable. Thus, a solution in case I exists with arbitrary constants.

II. Consider the case \(k=v=0\). The equations of the problem have the form

\[ \begin{gathered} \omega^3=0,\qquad \omega^0_3=0,\\ \omega^3_1=a\omega^1,\qquad \omega^0_1=c\sigma\omega^2,\\ \omega^3_2=c\omega^2,\qquad d\ln\sigma=-\omega^0_0+\omega^1_1-\omega^2_2+\omega^3_3,\\ \omega^2_1=0,\qquad a[\omega^1_3\omega^1]+[\omega^0_2\omega^2]=0,\\ \omega^1_2=0,\qquad [\omega^0_2\omega^1]-c[\omega^1_3\omega^2]=0. \end{gathered} \tag{15} \]

The system of quadratic equations has the form

\[ \begin{gathered} [\Delta\ln a\,\omega^1]=0,\qquad [\omega^0_2\omega^1]-c[\omega^1_3\omega^2]=0,\\ [\Delta\ln c\,\omega^2]=0,\qquad a[\omega^1_3\omega^1]+[\omega^0_2\omega^2]=0. \end{gathered} \]

Expanding by Cartan’s lemma, we obtain

\[ \begin{aligned} \Delta\ln a&=A_1\omega^1,\qquad &\omega^0_2&=aA_3\omega^1+cA_4\omega^2,\\ \Delta\ln c&=A_2\omega^2,\qquad &\omega^1_3&=-A_4\omega^1+A_3\omega^2. \end{aligned} \]

Here \(N=4\). Since \(s_1=4\) and \(q=4\), we have \(s_2=0\) and \(N=Q\). Consequently, system (15) is in involution. A solution exists with four arbitrary functions of one argument.

Received
22 III 1962

CITED LITERATURE

\(^{1}\) S. P. Finikov, Theory of Congruences, Moscow, 1950.
\(^{2}\) S. P. Finikov, Cartan’s Method of Exterior Forms, Moscow–Leningrad, 1948.
\(^{3}\) S. P. Finikov, Theory of Pairs of Congruences, Moscow, 1956.