PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 184
1970

DIFFERENTIAL GEOMETRY OF THE FAMILY \(K(2,6)\)

V. I. MATVEEVKO

(Presented by the Department of Higher Mathematics)

In the present article we continue the study of multiparameter families of second-order curves (conics) in three-dimensional projective space ([3]—[4]). We consider the family \(K(2,6)\)—a six-parameter family of nondegenerate conics \(C\), whose planes form a two-parameter family. This family of conics exists with an arbitrary function of six arguments. The notation and terminology correspond to those adopted in [1]—[5].

Place the vertex \(A_3\) of the moving frame \(\{A_\alpha\}\) at the characteristic point \(M\) of the plane of the conic (here the case in which the conic passes through the point \(M\) is excluded), the vertices \(A_i\) at the points of the conic polar-conjugate to it with respect to the point \(A_3\), and the vertex \(A_4\) at an arbitrary point of projective space not lying in the plane of the conic. Here and in what follows we agree to assume that
\(i,j,g,k,t=1,2;\ i\ne j;\ k\ne t;\ m,n,r=1,2,3;\ m\ne n;\ n\ne r;\ m\ne r;\)
\(\alpha,\beta,\gamma=1,2,3,4;\ \lambda=1,2,3,4,5,6;\ \sigma=i+2;\ \varkappa=j+2;\ \xi=i+4;\)
\(\eta=j+4;\) no summation over \(i,j,g,r,\alpha\)! The derivation formulas of the frame \(\{A_\alpha\}\) have the form:

\[ dA_\alpha=\omega_\alpha^\beta A_\beta, \tag{1} \]

where \(\omega_\alpha^\beta\) are Pfaffian forms satisfying the structure equations of projective space

\[ D\omega_\alpha^\beta=[\omega_\alpha^\gamma\omega_\gamma^\beta]. \tag{2} \]

The equations of the conic \(C\) with respect to this frame have the form:

\[ (x^3)^2-2px^1x^2=0,\quad x^4=0,\quad p\ne 0. \tag{3} \]

The forms

\[ \omega_m^4,\ \omega_i^j,\ \omega_3^i,\ \omega_i^3,\ \omega_k^k-2\omega_3^3-d\ln p \]

are principal forms. We choose as the basic forms

\[ \omega_i=\omega_i^4,\quad \omega_\sigma=\omega_i^j,\quad \omega_\xi=\omega_3^i-\frac{1}{p}\omega_i^3. \]

The remaining principal forms are expressed through them in the form:

\[ \omega_3^4=0,\quad \omega_3^i=C_3^{ik}\omega_k,\quad \omega_i^3=p\left(C_3^{jk}\omega_k-\omega_\xi\right), \]

\[ \omega_k^k-2\omega_3^3-d\ln p=C^\lambda\omega_\lambda, \tag{4} \]

where \(C_3^{12}=C_3^{21}\). Denote by \(D_\beta^\alpha\) the characteristic points of the conic \(C\) of the submanifold \(\Psi_1^\alpha\) of conics [5]. The submanifold \(\Psi_1^i\) \([\Psi_1^\sigma]\) of conics \(\omega_1=\omega_2=\omega_\lambda=\omega_5=\omega_6=0\) \([\omega_1=\omega_2=\omega_3=\omega_4=\omega_\eta=0]\) is characterized geometrically by the fact that for it the line \(A_1A_2\) \([A_iA_3]\) and the point \(A_j\) are fixed. By the choice of the basic forms \(\omega_\lambda\), the cases in which the dimension of the manifold of planes of conics is less than two, and the cases of fixed points \(A_i\) for the submanifolds \(\Psi_1^i\) and \(\Psi_1^\sigma\) of conics, are excluded.

From the system of equations (4), in the usual way [1], we obtain the following system of differential equations of the intrinsic fundamental object \(C_1\{C_3^{ik}, C^\lambda, p\}\):

\[ \delta C_3^{ig}=C_3^{ig}(\pi_3^3+\pi_4^4-\pi_i^i-\pi_g^g),\qquad \delta\ln p=\pi_k^k-2\pi_3^3, \]

\[ \delta C^i=C^i(\pi_4^4-\pi_i^i)-\pi_4^i+C^\sigma\pi_4^j-\frac{C^\xi}{p}\pi_4^3, \tag{5} \]

\[ \delta C^\sigma=C^\sigma(\pi_j^j-\pi_i^i),\qquad \delta C^\xi=C^\xi(\pi_j^j-\pi_3^3). \]

From these equations we conclude directly that the quantities \(p\), \(C_3^{ik}\), \(C^\sigma\), \(C^\xi\) are relative invariants. The condition \(C_3^{ij}=0\) characterizes a family of conics for which the net of lines \(\omega_1\omega_2=0\) on the surface \((A_3)\) is conjugate. The nonholonomic submanifold \(\Psi_5\) [5] of conics \(\omega_j=0\) is characterized geometrically by the fact that for it the line \(A_jA_3\) is a characteristic plane of the conic. A characteristic element of the plane [straight line] \(l\) of the submanifold \(\Psi_{\lambda-1}\) of planes [straight lines] is called such a straight line [point] whose displacement, for all submanifolds \(\Psi_1\) belonging to the submanifold \(\Psi_{\lambda-1}\), does not leave the plane [straight line] \(l\). The condition \(C_3^{ii}=0\) characterizes a family of conics for which the line \(\omega_i=0\) on the surface \((A_3)\) is an asymptotic line. For this family of conics the tangent to the line \(\omega_j=0\) on the surface \((A_3)\) coincides with the line \(A_3A_j\). The condition \(C^\sigma=0\) \([C^\xi=0]\) characterizes a family of conics for which the point \(A_j\) is a quadruple [triple] characteristic point of the conic \(C\) for the submanifold \(\Psi_1^i\) \([\Psi_1^\sigma]\) of conics. The geometric characteristics of the relative invariants given here follow from formulas (1) and the system of equations

\[ (x^3)^2-2px^1x^2=0,\quad x^4=0,\quad x^k\omega_k=0, \]

\[ \omega_3(x^1)^2+\omega_4(x^2)^2+C^\lambda\omega_\lambda x^1x^2+\omega_5x^1x^3+\omega_6x^2x^3=0 \tag{6} \]

for determining the points of intersection of the initial and adjacent conics of the family \(K(2,6)\).

The quantities

\[ A=\frac{C_3^{11}C_3^{22}}{(C_3^{12})^2},\qquad B_i=\frac{pC_3^{jj}}{C_3^{12}(C^i)^2},\qquad H_i=\frac{C_3^{12}}{C^\sigma C_3^{jj}} \]

are absolute invariants. Geometrically they are characterized through cross ratios of quadruples of points in the following way:

\[ A=(A_1A_2;\,S_3^2S_3^1),\qquad B_i=\frac12(A_iA_3;\,S_j^iP_i^j)^2, \]

\[ H_i=-(A_iA_j;\,P^iS_3^j). \tag{7} \]

where \(S_r^t\) is the projection from the point \(A_r\) onto the line \(A_mA_n\) of the point of intersection of the tangent to the line \((A_3)_{\omega_i=0}\) with the conic \(C\), \(P_i^t\) is the point of intersection of the line \(A_1A_2\) with the line \(D_3^iD_4^i\), \(P_j^\sigma\) is the point of intersection of the lines \(A_jA_3\) and \(A_iD_4^\sigma\), and \(P_i^\sigma\) is the point lying on the line \(A_iA_3\) and polar-conjugate to the point \(P_j^\sigma\) with respect to the conic \(C\). Using expressions (7), we obtain the following equalities:

\[ (A_1A_3;\, P_1^4P_1^3)=(A_2A_3;\, P_2^3P_2^4), \]

\[ (A_1A_2;\, S_3^2S_3^1)=(A_iA_3;\, S_j^iS_j^i). \]

Excluding from consideration the cases in which the relative invariants \(C_3^{12}, C^5\) vanish, we carry out the following normalization of the frame:

\[ p=C_3^{12}=C^5=1. \tag{8} \]

If one also takes into account the equiprojectivity condition \((A_1A_2A_3A_4)=1\), then we obtain \(\pi_\alpha^\alpha=0\). The forms \(\omega_\alpha^\alpha\) become principal, and their expansion in the basis forms may be written in the form

\[ \omega_\alpha^\alpha=C_\alpha^{\alpha\lambda}\omega_\lambda, \tag{9} \]

where \(C_\mu^{\beta\lambda}=0\).

From the system of equations (9), in the usual way we obtain the following system of differential equations of the internally prolonged fundamental object \(C_2\{C_1,C_r^\lambda\}\) [1]:

\[ \delta C_i^{li}=C_i^{l\delta}\pi_4^i-C_i^{li}\pi_4^3-\pi_4^l,\qquad \delta C_r^{r,\alpha+2}=0, \tag{10} \]

\[ \delta C_i^{lj}=C_i^{l\alpha}\pi_4^i-C_i^{l\eta}\pi_4^3,\qquad \delta C_3^{3l}=C_3^{3\sigma}\pi_4^j-C_3^{3\xi}\pi_4^3. \]

Since the systems of equations (5) and (10) are algebraically solvable with respect to all linearly independent forms \(\pi_\alpha^\beta=\pi_\alpha^\beta-\delta_\alpha^\beta\pi_4^4\), the object \(C_2\) is fundamental [1].

The systems of equations (5) and (10) make it possible to carry out the following normalization of the frame:

\[ S^1=S^2=1,\quad C_r^l+C_1^5+C_3^{11}(C_1^6-1)+C_1^3+S^3(C_1^5-1)=0,\quad C_i^l\ne0, \tag{11} \]

where

\[ C_i^\lambda=C_i^{i\lambda}-C_3^{3\lambda},\qquad S^i=\frac{1-C_j^i-C_3^{ll}C_j^\eta}{C_j^\xi},\qquad S^\sigma=-\frac{C_j^\sigma}{C_j^\xi}. \]

Under this normalization all secondary parameters are fixed and the frame becomes canonical. Let us clarify the geometric meaning of the normalization (11). The nonholonomic submanifold \(\Psi_2\) of conics

\[ \omega_\xi=\omega_\sigma=\omega_\varepsilon-C_3^{\eta j}\omega_j=\omega_\eta-S^j\omega_j-S^x\omega_x=0 \tag{12} \]

is characterized geometrically by the fact that for it the point \(A_l\) is fixed, while the point \(P_i^3\) is the characteristic point of the line \(P_i^3A_j\). The points \(A_j\) and \(P_j^3\) for this submanifold describe nonholonomic surfaces, whose tangent planes we denote by \(a_j\) and \(b_j\). Under the normalization (11) the vertex \(A_4\) of the frame \(\{A_\alpha\}\) is placed at the point of intersection of the planes \(a_1, a_2\), and \(b_1\). The inequality \(C_i^\eta\ne0\) excludes the case in which the point \(P_i^3\) is fixed for the submanifold (12).

The derivative formulas of the constructed canonical frame have the form:

\[ dA_i=C_i^{i\lambda}\omega_\lambda A_i+\omega_\sigma A_j+ (\omega_i+C_3^{j\nu}\omega_j-\omega_\xi)A_3+\omega_iA_4, \tag{13} \]

\[ dA_3=(C_3^{kk}\omega_k+\omega_l)A_k+C_3^{3\lambda}\omega_\lambda A_3, \qquad dA_4=C_4^{\beta\lambda}\omega_\lambda A_\beta . \]

The structure equations (2) lead to the following system of finite relations:

\[ 2C_k^{k\sigma}+C_3^{ii}=C_k^{\lambda\xi}=0. \tag{14} \]

A nonholonomic submanifold \(\Psi_5\) of the conic \(\Psi_{\alpha}^{\rho}=0\) is characterized geometrically by the fact that for it the point \(A_\alpha\) is the characteristic point of the plane \(A_\alpha A_\rho A_\mu\) \((\alpha,\rho,\tau,\mu)=1,2,3,4;\ \alpha\ne\rho,\tau,\mu;\ \rho\ne\tau,\mu;\ \tau\ne\mu\). A nonholonomic submanifold \(\Psi_5^r\) of the conic \(\omega_\xi=0\) is characterized geometrically by the fact that for it the points \(A_i\) and \(A_3\) are polar conjugates with respect to the projection of the adjacent conic from the point \(A_4\) onto the plane of the initial conic \(C\).

From the system of equations (6) and formulas (13) the following propositions follow:

1. The points \(A_i\) are double foci of the conic \(C\) of the nonholonomic congruence of conics \(\omega_3=\omega_4=\omega_5=\omega_6=0\), and to these foci there correspond the focal families \(\omega_i=0\). The other two foci of the conic \(C\) of this congruence lie on the line passing through the point \(A_3\), and to them there corresponds the double focal family \(C^k\omega_k=0\).

2. For the nonholonomic congruence of conics \(\omega_j=\omega_x=\omega_5=\omega_6=0\) \([\omega_j=\omega_3=\omega_4=\omega_\eta=0]\), the foci of the conic \(C\) coincide with the points \(D_\xi^i[D_\xi^j]\). Moreover, the focus \(A_j\) is a quadruple focus, and to it there corresponds an indeterminate focal family. To the remaining two foci of the conic \(C\) of this congruence there corresponds the double focal family \(\omega_i=0\). Moreover, the focal lines \(\omega_i=0\) are plane lines situated in the planes of the conics \(C\).

3. For the nonholonomic congruence of conics \(\omega_j=\omega_\sigma=\omega_5=\omega_6=0\), the foci of the conic \(C\) are the points \(D_\beta^j\) and the point \(A_j\). Moreover, the points \(A_k\) are double foci. To the focus \(A_j\) there corresponds the focal family \(\omega_x=0\), and to the remaining foci—the focal family \(\omega_i=0\). Moreover, the focal lines \(\omega_i=0\) are plane lines situated in the planes of the conics \(C\).

4. For the nonholonomic congruence of conics \(\omega_j=\omega_3=\omega_4=\omega_\xi=0\), the foci of the conic \(C\) coincide with the points \(D_\beta^x\). Moreover, the point \(A_j\) is a triple focus, and the point \(A_i\) is a double focus. To the focus \(A_j\) there corresponds an indeterminate focal family, and to the remaining foci—the focal family \(\omega_i=0\). Moreover, the focal lines \(\omega_i=0\) are plane lines situated in the planes of the conics \(C\).

5. The foci of the ray of the nonholonomic congruence of lines

\[ (A_1A_2)_{\omega_3=\omega_4=\omega_5=\omega_6=0} \]

divide the points \(A_1\) and \(A_2\) harmonically.

REFERENCES

1. G. F. Laptev. Differential geometry of immersed manifolds. Trudy Moskovskogo matematicheskogo obshchestva, 2, 1953, 275–382.

2. S. P. Finikov. The Method of Exterior Forms of Cartan. GITL, Moscow–Leningrad, 1948.

3. V. S. Malakhovskii. Nondegenerate congruences of curves of second order in three-dimensional projective space. Geometricheskii sbornik, issue 3 (Trudy Tomskogo universiteta, 168), 1963, 43–53.

4. V. I. Matveenko. Multiparameter families of curves of second order in three-dimensional projective space. Twenty-fifth Scientific-Pedagogical Conference of the Mathematics Departments of Higher Educational Institutions of the Urals Zone. Abstracts of reports and communications, Sverdlovsk, 1967, 59–60.

5. R. N. Shcherbakov. On the method of the moving frame and frames of submanifolds. Geometricheskii sbornik, issue 6 (Trudy Tomskogo universiteta, 191), 1967, 179–194.