G. B. Khasin

\(R\)-Nets with Foliating Congruences of Axes

(Presented by Academician P. S. Novikov on 19 XI 1962)

In proving the existence in three-dimensional projective space of such nets that the congruences of their first and second axes are foliated respectively with the congruences of the first and second axes of the Laplace transforms of these nets, two different cases arise \((^{1})\). With a suitable choice of frame, one of these classes of nets is described by the system:

\[ \begin{gathered} \omega^{3}=0,\qquad \omega_{2}^{1}=0,\qquad d\ln\sigma=\omega_{1}^{1}+\omega_{3}^{3}-\omega_{0}^{0}-\omega_{2}^{2},\\ \omega_{1}^{3}=a\omega^{1},\qquad \omega_{3}^{0}=0,\qquad [\omega_{2}^{0}\omega^{1}]-c[\omega_{3}^{1}\omega^{2}]=0,\quad (a)\\ \omega_{2}^{3}=c\omega^{2},\qquad \omega_{3}^{2}=\sigma\omega^{1},\qquad [\omega_{2}^{0}\omega^{2}]+a[\omega_{3}^{1}\omega^{1}]=0.\quad (b)\\ \omega_{1}^{2}=0,\qquad \omega_{1}^{0}=c\sigma\omega^{2}, \end{gathered} \tag{1} \]

In the present work the geometric properties of these nets are considered.

Theorem 1. In order that a given net be a net described by system (1), it is necessary and sufficient that it be an \(R\)-net whose axes are the axes of at least one of its Laplace transforms.

Proof. If \(M_{0}\) is the given net, \(M_{1}\) and \(M_{-1}\) are its Laplace transforms, \(N_{2}\) is the Laplace transform of the net \(M_{1}\), \(M_{3}\) is the point of intersection of the ray \([M_{1}N_{2}]\) with the plane tangent to \(M_{-1}\), and \(M_{2}\) is the point of intersection of the ray \([M_{0}M_{-1}]\) with the plane tangent to \(N_{2}\), then by virtue of system (1) \(M_{-1}\equiv M_{2}\) and \(M_{3}\equiv N_{2}\). Thus, the axes of the net \(M_{0}\) are the axes of the net \(M_{1}\). From equations (1a) and (1b) it follows that

\[ \{\omega_{2}^{0}=aA_{3}\omega^{1}+cA_{4}\omega_{2},\quad \omega_{2}^{1}=-A_{4}\omega^{1}+A_{3}\omega^{2}\}. \tag{2} \]

From system (2) it follows that the asymptotic forms of the nets \(M_{0}\), \(M_{1}\), and \(M_{2}\) are proportional, which proves the necessity of the assertion. If \(M_{0}\) is an \(R\)-net whose axes coincide with the axes of the net \(M_{1}\), then \(M_{3}\equiv N_{2}\) and \(M_{-1}\equiv M_{2}\), and, choosing \([M_{0}M_{1}M_{2}M_{3}]\) as the frame, we obtain:

\[ \{\omega^{3}=0,\ \omega_{1}^{2}=0,\ \omega_{2}^{1}=0,\ \omega_{1}^{3}=a\omega^{1},\ \omega_{2}^{3}=c\omega^{2}\}. \tag{3} \]

Differentiating these equations and expanding by Cartan’s lemma, we obtain

\[ \begin{aligned} \omega_{1}^{0}&=\beta\omega^{2}+aA\omega^{1},\qquad &\omega_{2}^{0}&=m\omega^{1}+cB\omega^{2},\\ \omega_{3}^{2}&=\sigma\omega^{1}-A\omega^{2},\qquad &\omega_{3}^{1}&=-B\omega^{1}+l\omega^{2}, \end{aligned} \]

whence the asymptotic forms of the nets \(M_{0}\), \(M_{1}\), and \(M_{2}\) have the form:
\[ \Phi_{0}=a(\omega^{1})^{2}+c(\omega^{2})^{2},\quad \Phi_{1}=a\sigma(\omega^{1})^{2}+\beta(\omega^{2})^{2},\quad \Phi_{2}=m(\omega^{1})^{2}+cl(\omega^{2})^{2}. \]
Since these forms are proportional, \(\beta=c\sigma,\ m=al\). These equalities turn into identities the equations of foliation of the congruences \([M_{0}M_{3}]\) and \([M_{1}M_{2}]\), whence the theorem follows.

Consider the special case when in system (2) \(A_{4}=0\). Joining equations (2), under the condition \(A_{4}=0\), to system (1), and differentiating the resulting equations exteriorly, we shall have:

\[ \begin{gathered} [\Delta\ln a;\omega^{1}]=0,\qquad \Delta\ln a=d\ln a+\omega_{0}^{0}-2\omega_{1}^{1}+\omega_{3}^{3},\\ [\Delta\ln c;\omega^{2}]=0,\qquad \Delta\ln c=d\ln c+\omega_{0}^{0}-2\omega_{2}^{2}+\omega_{3}^{3},\\ d\ln\sigma=\omega_{1}^{1}+\omega_{3}^{3}-\omega_{0}^{0}-\omega_{2}^{2},\\ d\ln A_{3}=\omega_{3}^{3}+\omega_{2}^{2}-\omega_{1}^{1}-\omega_{0}^{0}. \end{gathered} \]

Adjoining the last two equations to system (1) and differentiating exteriorly, we shall have: \([\Delta \ln a; \omega^1]=0,\ [\Delta \ln c; \omega^2]=0\), whence \(N=Q=S_1=2\). Consequently, the case \(A_4=0\) exists with arbitrariness of two functions of one argument.

Theorem 2. In order that the net \(M_0\) belong to the case \(A_4=0\), it is necessary and sufficient that the nets \(M_0, M_1, M_2, M_3\) form a closed quadruple of \(R\)-nets.

Proof. From equations (1) and (2), under the condition \(A_4=0\), it follows that
\(dM_3=\sigma \omega^1 M_2+\omega_3^3 M_3 \pmod{\omega^2=0}\). Similarly,
\(dM_2=\omega_2^2 M_2+c\omega^2 M_3 \pmod{\omega^1=0}\). The converse is also true: every closed quadruple of \(R\)-nets satisfies the case \(A_4=0\).

From the equation \([dF,F,M_0,M_3]=0\) it follows that the foci of the congruence \([M_0M_3]\) have the form
\(F_1=\sqrt{A_3\sigma}\,M_0+M_3,\ F_2=-\sqrt{A_3\sigma}\,M_0+M_3\), whence
\((F_1F_2;M_0M_3)=-1\). In this case the equation of the return edge \(F_1\) has the form
\(\sqrt{\sigma}\omega^1+\sqrt{A_3}\omega^2=0\), and the return edge \(F_2\) has the form
\(\sqrt{\sigma}\omega^1-\sqrt{A_3}\omega^2=0\). The requirement that they be conjugate with respect to the form \(\Phi_0\) leads to the condition \(aA_3=c\sigma\), which is equivalent to the requirement

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

Let \(\omega^1=\alpha_1du,\ \omega^2=\alpha_2dv,\ \omega_0^0=\alpha_3du+\alpha_4dv,\ \omega_1^1=\alpha_5du+\alpha_6dv\). Then from
\(dM_0=\omega_0^0M_0+\omega^1M_1+\omega^2M_2\) and
\(dM_1=\omega_1^0M_0+\omega_1^1M_1+\omega_1^3M_3\) we have
\(M_{0u}=\alpha_3M_0+\alpha_1M_1\) and
\(M_{1v}=\beta\alpha_2M_0+\alpha_6M_1\), whence we have
\(M_{0uv}=\{\alpha_6+(\ln \alpha_1)_v\}M_{0u}+\alpha_3M_{0v}+\{\alpha_{3v}+\beta\alpha_1\alpha_2-\alpha_3(\ln \alpha_1)_v-\alpha_3\alpha_5\}M_0\).

The Darboux invariants are written as \((^2)\):

\[ h=\alpha_{3v}+\beta\alpha_1\alpha_2-\alpha_{6u}-(\ln \alpha_1)_{uv},\qquad j=\beta\alpha_1\alpha_2. \]

Consequently,
\(h-j=\alpha_{3v}-\alpha_{6u}-(\ln \alpha_1)_{uv}\). From the equality
\(D\omega^1=[\omega_0^0-\omega_1^1\ \omega^1]\) we obtain
\((\ln \alpha_1)_v=\alpha_4-\alpha_6\), consequently,
\((\ln \alpha_1)_{uv}=\alpha_{4u}-\alpha_{6u}\). Conditions \((*)\) give
\(D\omega_0^0=0\), i.e. \(\alpha_{3v}=\alpha_{4u}\). Consequently,
\((\ln \alpha_1)_{uv}=\alpha_{3v}-\alpha_{6u}\) and \(h=j\). But if \(M_0\) is a net with equal invariants, then \(M_3\) is also a net with equal invariants \((^2)\). From equation \((*)\) it follows that the net \(M_1\), and hence also \(M_2\), is also a net with equal invariants. Analogous assertions are true also for the congruence \([M_1M_2]\). The foci of the ray \([M_1M_2]\) have the form:
\(F_0=\sqrt{A_3}M_1+\sqrt{\sigma}M_2;\ F_3=\sqrt{A_3}M_1-\sqrt{\sigma}M_2;\)
the equation of the return edge \(F_0\) is
\(a\sqrt{A_3}\omega^1+c\sqrt{\sigma}\omega^2=0\), the equation of the return edge \(F_3\) is
\(a\sqrt{A_3}\omega^1-c\sqrt{\sigma}\omega^2=0\). If \(M_0\) is a net with equal invariants, then the return edge \(F_0\) corresponds to the return edge \(F_1\), and the return edge \(F_2\) corresponds to the return edge \(F_3\). Consequently, in this case the net \(M_0\) is harmonic to the congruence \([M_1M_2]\). From what has been said it follows

Theorem 3. In the case \(A_4=0\), the foci of the congruence \([M_0M_3]\) harmonically separate the points \(M_0\) and \(M_3\). If \(M_0\) is a net with equal Darboux invariants, then the net \(M_0\) is conjugate to the congruence \([M_0M_3]\) and harmonic to the congruence \([M_1M_2]\). Analogous assertions are true for the net \(M_1\).

The equations of the net \(M_0\) with equal invariants in the case \(A_4=0\) have the form:

\[ \begin{array}{llll} \omega^3=0, & \text{(a)} & \omega_1^2=0, & \text{(г)} \qquad \omega_3^2=\sigma\omega^1, \quad \text{(ж)}\\ \omega_1^3=a\omega^1, & \text{(б)} & \omega_2^1=0, & \text{(д)} \qquad \omega_1^0=c\sigma\omega^2, \quad \text{(з)}\\ \omega_2^3=c\omega^1, & \text{(в)} & \omega_3^0=0, & \text{(е)} \qquad \omega_2^0=c\sigma\omega^1, \quad \text{(и)}\\ a\omega_3^1=c\sigma\omega^2, & \text{(к)} & d\ln\sigma=\omega_0^1+\omega_3^3-\omega_0^0-\omega_2^2. & \text{(л)} \end{array} \tag{4} \]

Differentiating (4б), (4в), (4и), and (4к), we shall have

\[ d\ln a=2\omega_1^1-\omega_0^0-\omega_3^3,\qquad d\ln c=2\omega_2^2-\omega_0^0-\omega_3^3. \]

These equations must be adjoined to system (4), and then the new system will prove to be completely integrable.

From the expressions for \(F_1, F_2, F_3, F_0\) we obtain

\[ [dF_1,F_1,F_0]=[dF_0,F_1,F_0]=0. \]

Consequently, the line \([F_0F_1]\) is fixed; analogously, the line \([F_2F_3]\) is fixed. Thus the axes of the net \(M_0\) belong to one linear congruence, whose directrices are \(F_0F_1\) and \(F_2F_3\). Under displacement along the line \(\sqrt{\sigma}\omega^1+\sqrt{A_3}\omega^2=0\), all first axes of the net \(M_0\) pass through \(F_1\), while under displacement along the line \(\sqrt{\sigma}\omega^1-\sqrt{A_3}\omega^2=0\), all first axes of the net \(M_0\) pass through \(F_2\). Similarly, under displacement along the line \(a\sqrt{A_3}\omega^1+c\sqrt{\sigma}\omega^2=0\), all second axes of the net pass through \(F_0\), while under displacement along the line \(a\sqrt{A_3}\omega^1-c\sqrt{\sigma}\omega^2=0\), all second axes of the net \(M_0\) pass through the point \(F_3\). The first axes pass through \(F_1(F_2)\) and the second axes pass through \(F_0(F_3)\) simultaneously if and only if the net \(M_0\) is a net with equal invariants.

Differentiating equations (2) externally, we obtain

\[ \Delta A_4=B_1\omega^1+B_2\omega^2,\qquad \Delta A_4=dA_4-A_4(\omega_3^3-\omega_0^0), \]

\[ a\Delta A_3=-aB_2\omega^1+cB_1\omega^2,\qquad \Delta A_3=dA_3+A_3(\omega_0^0+\omega_1^1-\omega_2^2-\omega_3^3). \]

Putting \(B_1=B_2=0\), we shall have:

\[ d\ln A_3=\omega_2^2+\omega_3^3-\omega_0^0-\omega_1^1,\qquad d\ln A_4=\omega_3^3-\omega_0^0. \]

Adjoining these equations to systems (1) and (2), we obtain a system in involution, and the arbitrariness of the solution depends on two functions of one argument.

Theorem 4. If \(B_1=B_2=0\), then all Laplace transforms of the net with even indices lie on the ray \([M_0M_3]\), and the transforms with odd indices lie on the ray \([M_1M_2]\).

Proof. Denote \(M_0=L_0,\ M_1=L_1,\ M_2=L_{-1},\ M_3=L_2\). Then

\[ dL_{-1}\pmod{\omega^1=0}=A_4M_0+cM_3=L_{-2}, \]

\[ dL_{-2}\pmod{\omega^2=0}=\omega_3^2L_{-1}+\omega_3^3L_{-2}, \]

whence it follows that \(L_{-2}\) is the focus of the ray \([L_{-1}L_{-2}]\). If we denote \(N_0=L_{-1},\ N_1=L_0,\ N_2=L_2,\ N_3=L_1\), then the forms of infinitesimal displacements of the frame \([N_0N_1N_2N_3]\), \(\theta_i^k\), satisfy equations analogous to systems (1), (2), and (6); here \(\bar a=1/aA_3,\ \bar c=A_3/c,\ \bar\sigma=1/A_3,\ \bar A_3=\sigma,\ \bar A_4=A_4/A_3\). Thus the assertion of the theorem is true for the next net, since it is true for the preceding one. The forms \(\pi_i^k\) of the infinitesimal displacements of the frame \([L_1L_2L_0L_3]\) satisfy a system of equations analogous to equations (1), (2), and (6). In this case \(\bar a=1/a,\ \bar c=1/c\sigma^2,\ \bar\sigma=\sigma A_3,\ \bar A_3=1,\ \bar A_4=A_4\), whence the validity of the theorem follows.

The equation of the foci of the ray \([M_0M_3]\) gives \(F_1=\lambda_1M_0+M_3,\ F_2=\lambda_2M_0+M_3\), where \(\lambda_1\) and \(\lambda_2\) are the roots of the equation \(\lambda^2-A_4\lambda-A_3\sigma=0\). The equations of the developable surfaces are \(\sigma\omega^1+\lambda_1\omega^2=0\) and \(\sigma\omega^1+\lambda_2\omega^2=0\). Similarly, the foci \(F_0\) and \(F_3\) of the ray \([M_1M_2]\) are written in the form \(F_{0(3)}=\mu_{1(2)}M_1+M_2\), where \(\mu_1\) and \(\mu_2\) are the roots of the equation \(\sigma\mu^2+A_4\mu-A_3=0\). The equation of the return edge \(F_0\) is \(a\mu_1\omega^1+c\omega^2=0\), the equation of the return edge \(F_3\) is \(a\mu_2\omega^1+c\omega^2=0\). If we require that the lines \(\sigma\omega^1+\lambda_1\omega^2=0\) and \(\sigma\omega^2+\lambda_2\omega^2=0\) be conjugate on the surface \(M_0\), then \(aA_3=c\sigma\), which means equality of the invariants of the net \(M_0\). In this case the lines \(\sigma\omega^1+\lambda_1\omega^2=0\) and \(a\mu_1\omega^1+c\omega^2=0\) coincide, whence it follows

Theorem 5. The net \(M_0\) is conjugate to the congruence \([M_0M_3]\) and to the harmonic congruence \([M_1M_2]\) if and only if \(M_0\) is a net with equal invariants.

The nets described in the condition of Theorem 5 satisfy a completely integrable system.

The expression \(A_4^2 + 4A_3\sigma\) is a relative invariant of the net \(M_0\), since
\[ d(A_4^2 + 4A_3\sigma)=2(\omega_3^3-\omega_0^0)(A_4^2+4A_3\sigma). \]
Its vanishing at \(0\) means that the congruences of axes of the nets \(M_0\) and \(M_1\) are parabolic. From the expressions for \(F_1,F_2,F_3,F_0\) we have
\[ dF_1=\omega_3^3F_1+\theta F_0, \]
where \(\theta\) is a form proportional to the form \(\sigma\omega^1+\lambda\omega^2\), and
\[ dF_0=\omega_2^2F_0+\nu F_1, \]
where \(\nu\) is a form proportional to the form \(a\mu_1\omega^1+c\omega^2\), whence it follows

Theorem 6. The axes of the net \(M_0\) belong to one and the same linear congruence, whose directrices are \(F_1F_0\) and \(F_2F_3\). The developable surfaces of the congruence of the first and second axes degenerate into cones with vertices at \(F_0,F_1,F_2,F_3\), whose equations are:
\[ a\mu_1\omega^1+c\omega^2=0,\qquad \sigma\omega^1+\lambda_1\omega^2=0,\qquad a\mu_2\omega^1+c\omega^2=0,\qquad \sigma\omega^1+\lambda_1\omega^2=0. \]

Moscow State University
named after M. V. Lomonosov

Received
19 XI 1962

REFERENCES

1. G. B. Khasin, DAN, 145, No. 6 (1962).
2. S. P. Finikov, Theory of Congruences, Moscow, 1950, p. 261.