SIX-PARAMETER FAMILY OF CONICS IN \(P_3\)

V. I. Matveenko

(Presented by the Department of Higher Mathematics)

In the present article the study of six-parameter families of curves of the second order (conics) in three-dimensional projective space is continued (see [4]). A family \(K(3,6)\) is considered—a six-parameter family of nondegenerate conics \(C\), whose planes form a three-parameter family. The principal object [1] is obtained. Relative and absolute invariants are found and geometrically characterized. A canonical frame is constructed; certain submanifolds of the family \(K(3,6)\), certain geometric images associated with the family \(K(3,6)\), and certain classes of this family are considered. The notation and terminology correspond to those adopted in [1]—[5]. The indices take the following values: \(i,j,k,g,t=1,2;\ i\ne j;\ k\ne t;\ m,n,r=1,2,3;\ m\ne n;\ m\ne r;\ n\ne r;\ \alpha,\beta,\gamma=1,2,3,4;\ \lambda=1,2,3,4,5,6;\ \sigma=i+3;\ \chi=j+3\). No summation over \(i,j,g,r,\alpha\)!

Place the vertices \(A_1\) and \(A_2\) of the moving frame \(\{A_\alpha\}\) at distinct points of the conic \(C\); place the vertex \(A_3\) at the pole of the line \(A_1A_2\) with respect to the conic \(C\), and the vertex \(A_4\) at an arbitrary point of the projective space not lying in the plane of the conic. 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:

\[ (x^3)^2-2px^1x^2=0,\quad x^4=0,\quad p\ne0. \tag{3} \]

We perform a partial normalization of the vertices of the frame \(\{A_\alpha\}\), setting \(p=1\). The forms

\[ \omega_n^4,\ \omega_i^j,\ \omega_3^j-\omega_i^3,\ \omega_k^k-2\omega_3^3 \]

are principal forms. We choose from them as basic forms

\[ \omega_n=\omega_n^4,\quad \omega_\sigma=\omega_i^j,\quad \omega_6=\omega_k^k-2\omega_3^3. \]

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

\[ \omega_3^j-\omega_i^3=C_i^\lambda\omega_\lambda. \tag{4} \]

From these equations, in the usual way [1], one obtains the following system of differential equations of the internal fundamental object \(C_1\{C_i^\lambda\}\):

\[ \begin{aligned} \delta C_i^g={}& C_i^g(\pi_3^3+\pi_4^4-\pi_g^g-\pi_j^j)+C_i^{g+3}\pi_4^{3-g} -C_i^3\pi_3^g+C_i^6\pi_4^g-C_i^{k+3}C_k^g\pi_k^3 \\ &-3C_i^6\sum_k C_t^g\pi_k^3+\delta_i^g\pi_4^3,\\ \delta C_i^3={}& C_i^3(\pi_4^4-\pi_j^j)+2C_i^6\pi_4^3-\pi_4^j -\sum_k (C_i^k+C_i^{k+3}C_k^3)\pi_k^3 -3C_i^6\sum_k C_t^3\pi_k^3,\\ \delta C_i^{g+3}={}& C_i^{g+3}\bigl[\pi_3^3-\pi_i^i+2\delta_j^g(\pi_i^i-2\pi_j^j)\bigr] -\sum_k C_i^{k+3}C_k^{g+3}\pi_k^3 \\ &-3C_i^6\sum_k C_t^{g+3}\pi_k^3+\delta_i^g(\pi_3^i+\pi_j^3),\\ \delta C_i^6={}& C_i^6(\pi_3^3-\pi_j^j)-\sum_k C_k^3C_i^{k+3}\pi_k^3+\pi_i^3 -3C_i^6\sum_k C_t^6\pi_k^3 . \end{aligned} \tag{5} \]

Since this system of equations is algebraically resolvable with respect to all independent secondary forms \(\pi_\alpha^\beta=\pi_2^\beta-\delta_\alpha^\beta\pi_4^4\), the object \(C_1\) is a principal one [1]. The system of equations (5) makes it possible to carry out the following normalization of the frame:

\[ C_1^5=C_2^6=0 . \tag{6} \]

The forms \(\omega_i^3\) and \(\omega_3^i\) become principal, and their expansion in the basic forms may be written in the form

\[ \omega_i^3=C_i^{3\lambda}\omega_\lambda,\qquad \omega_3^i=C_3^{i\lambda}\omega_\lambda . \tag{7} \]

Let us clarify the geometric meaning of the normalizations (6). A holonomic complex of conics \(\omega_1=\omega_2=\omega_3=0\) is geometrically characterized by the fact that, for it, the plane of the conic is fixed. Among the submanifolds \(\Psi_1\) [5] of conics

\[ \omega_1=\omega_2=\omega_3=0,\quad \omega_4:\omega_5:\omega_6=b_1:b_2:b_3, \tag{8} \]

belonging to this complex, in the general case there exist forty-eight such submanifolds \(\Psi_1\) of conics, for each of which the conic \(C\) has two double characteristic points \(M_u^t\) \((u=1,2,3,\ldots,48)\). This proposition follows from considering the system of equations

\[ (x^3)^2-2x^1x^2=0,\qquad x^4=0,\qquad x^n\omega_n=0, \]

\[ x^1x^2\omega_6+\sum_k\bigl[(x^k)^2\omega_{k+3}+x^kx^3(C_3^{t\lambda}-C_k^{3\lambda})\omega_\lambda\bigr]=0 \tag{9} \]

for determining the intersection points of the initial and neighboring conics of the family \(K(3,6)\). Under the normalization (6), the vertices \(A_i\) of the frame \(\{A_\alpha\}\) are placed at the points \(M_1^i\). The submanifold \(\Psi_3^i\) of conics \(\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=0\) is geometrically characterized by the fact that, for it, the plane of the conic is fixed, and the points \(A_i\) are double characteristic points of the conic \(C\). Denote by \(C_i^*\) the conic passing through the point \(A_3\) and the characteristic points of the conic \(C\) corresponding to the submanifold \(\Psi_1^i\) of conics \(\omega_1=\omega_2=\omega_3=\omega_4=\omega_6=0\). The submanifold \(\Psi_1^i\) of conics is geometrically characterized by the fact that, for it, the plane of the conic is fixed, the points \(A_1\)

and \(A_2\) are polar-conjugate with respect to the additional conic \(C_i^*\), while the point \(A_j\) is a point of the envelope of the family of lines \((A_jA_3)\). Using the system of equations (9), we obtain the equations of the conic \(C_i^*\)

\[ (x^i)^2+C_k^\sigma x^k x^3=0,\qquad x^4=0. \tag{10} \]

It follows from these equations that the conic \(C_i^*\) passes through the point \(A_j\) and, in addition to the point \(A_3\), intersects the line \(A_iA_3\) also at the point \(D_i=C_i^\sigma A_i-A_3\). By the choice of the base forms \(\omega_\lambda\), the cases are excluded when the dimension of the variety of planes of the conics is less than three, when the characteristic points of the conic \(C\) of the subvariety \(\Psi_1^3\) become indeterminate, and the cases in which the conics \(C_i^*\) pass through the points \(A_i\).

From the system of equations (7), in the usual way, one obtains the following system of differential equations of the internally prolonged fundamental object \(C_2\{C_1,C_i^{3\lambda},C_3^{i\lambda}\}\) [1]:

\[ \begin{aligned} \delta C_i^{3g} &= C_i^{3g}(\pi_4^4-\pi_3^3+\pi_i^i-\pi_g^g) +C_i^{3,g+3}\pi_4^{3-g}+C_i^{36}\pi_4^g-\delta_i^g\pi_4^3,\\ \delta C_i^{33} &= C_i^{33}(\pi_i^i-2\pi_3^3+\pi_4^4)+2C_i^{36}\pi_4^3,\\ \delta C_3^{ig} &= C_3^{ig}(\pi_3^3+\pi_4^4-\pi_i^i-\pi_g^g) +C_3^{i,g+3}\pi_4^{3-g}+C_3^{i6}\pi_4^j,\\ \delta C_3^{i3} &= C_3^{i3}(\pi_4^4-\pi_i^i)+2C_3^{i6}\pi_4^3-\pi_4^i,\\ \delta C_3^{i,g+3} &= C_3^{i,g+3}(\pi_3^3-2\pi_g^g-\pi_j^j), &\qquad \delta C_3^{i6} &= C_3^{i6}(\pi_3^3-\pi_i^i),\\ \delta C_i^{3,g+3} &= C_i^{3,g+3}(2\pi_{3-g}^{3-g}-\pi_j^j-\pi_3^3), &\qquad \delta C_i^{36} &= C_i^{36}(\pi_i^i-\pi_3^3). \end{aligned} \tag{11} \]

From the systems of equations (5) and (11) we immediately observe that the quantities \(C_k^\sigma,\ C_3^{i,n+3},\ C_i^{36},\ C_k^{3\sigma}\) are relative invariants. The condition \(C_i^\sigma=0\) characterizes the family of conics for which the points \(A_i\) and \(A_3\) are polar-conjugate with respect to the conic \(C_i\). The condition \(C_j^\sigma=0\) characterizes the family of conics for which the point \(A_i\) is a double characteristic point of the conic \(C\) of the subvariety \(\Psi_1^i\). The condition \(C_3^{i,n+3}=0\) characterizes the family of conics for which the tangent \(l_3^{\,i,n+3}\) to the line \((A_3)_{\omega_1=\omega_2=\omega_3=\omega_{n+3}=\omega_{r+3}=0}\) coincides with the line \(A_3A_j\). The condition \(C_i^{36}=0\) \([C_i^{3x}=0]\) characterizes the family of conics for which the point \(A_i\) is fixed for the subvariety \(\Psi_1^3\) \([\Psi_1^i]\). The condition \(C_i^{3\sigma}=0\) characterizes the family of conics for which the tangent \(l_i^\sigma\) to the line \((A_i)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=0}\) coincides with the line \(A_1A_2\).

The quantities

\[ A=\frac{C_1^{36}C_2^4}{C_2^{36}C_1^4},\qquad B_i=C_j^\sigma C_3^{ig},\qquad E_i=\frac{C_3^{ix}}{C_j^{3x}}, \]

\[ H_i=C_j^{3x}C_i^\sigma,\qquad G_i=\frac{C_i^\sigma C_3^{i\sigma}}{C_2^\sigma C_3^{2\sigma}} \]

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

\[ \begin{gathered} A=(A_1A_2;\,M_{12}^6S_{13}),\qquad B_i=(A_iA_3;\,S_jM_{i3}^{\sigma}),\\ E_i=1-(A_iA_3;\,S_jQ_j),\qquad H_i=-2(A_iA_3;\,Q_jS^i),\\ G_i=-(A_1A_2;\,S_{i3}Q_{31}). \end{gathered} \tag{12} \]

where \(S_{13}\) is the point of intersection of the line \(A_1A_2\) with the polar of the point \(A_3\) with respect to the conic \(C_i\), \(S_i\) is the point of intersection of the line \(A_jA_3\) with the polar of the point \(D_i\) with respect to the conic \(C\), \(M_{12}^6[M_{i3}^i]\) is a point of the envelope of the family of lines
\[ (A_1A_2)_{\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=0} \quad \left[(A_iA_3)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=0}\right], \]
\(Q_{3m}[Q_i]\) is the point of intersection of the line \(A_1A_2\) \([A_jA_3]\) with the tangent to the line
\[ (A_3)_{\omega_1=\omega_2=\omega_3=\omega_{n+3}=\omega'_{r+3}=0} \quad \left[(A_i)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=0}\right], \]
\(S^i\) is the point of intersection of the line \(A_iA_3\) with the polar of the point \(A_i\) with respect to the conic \(C_i^*\).

Using the expressions (12), we obtain the following equalities:
\[ (A_1A_2;\, M_{12}^6Q_{33})=-1,\qquad (A_iA_3;\, S^iD_i)=2, \]
\[ (A_1A_3;\, S_2M_{13}^4)=(A_1A_3;\, Q_2D_1) =(A_2A_3;\, S_1M_{23}^5)=(A_2A_3;\, Q_1D_2). \]

Excluding from consideration the cases in which the relative invariants \(C_i^{36}\) vanish, we carry out the following fixing of the frame:
\[ C_j^{36}C_i^{3i}-C_i^{36}C_j^{3i} = C_3^{16}C_3^{23}-C_3^{26}C_3^{13}=0, \qquad C_3^{ij}C_i^{36}-C_3^{i6}C_i^{3j}=1, \tag{13} \]
\[ C_i^{36}\ne 0,\qquad \sum_k C_k^{36}C_3^{t6}\bigl(C_t^{36}C_k^{3,k+3}-C_k^{36}C_t^{3,k+3}\bigr)\ne 0. \]

Taking also into account the condition of equiprojectivity \((A_1A_2A_3A_4)=1\), we obtain that all secondary parameters are fixed and the frame \(\{A_\alpha\}\) becomes canonical. Let us clarify the geometric meaning of the fixing (13). Denote by
\[ a_i=\{A_i,\; C_j^{36}A_j+(C_j^{36}C_i^{3r}-C_i^{36}C_j^{3r})A_3, \tag{14} \]
\[ (C_j^{36}C_i^{3i}-C_i^{36}C_j^{3i})A_3+C_j^{36}A_4\} \]
the tangent plane to the nonholonomic surface
\[ (A_i)_{\omega_j=\omega_3=\omega_x=\omega_j^3=0}, \]
and by
\[ a_3=\{A_3,\; C_3^{k6}A_k,\; C_3^{k3}A_k+A_4\} \tag{15} \]
the tangent plane to the nonholonomic surface
\[ (A_3)_{\omega_1=\omega_2=\omega_4=\omega_5=0}. \]

The nonholonomic submanifold \(\Psi_2\) [5] of conics
\[ \omega_j=\omega_3=\omega_x=\omega_j^3=0 \]
is geometrically characterized by the fact that for it the point \(A_j\) is fixed, while the line \(A_iA_3\) is the characteristic of the plane of the conic. The nonholonomic submanifold \(\Psi_2\) of conics
\[ \omega_1=\omega_2=\omega_4=\omega_5=0 \]
is geometrically characterized by the fact that for it the points \(A_i\) are the characteristic points of the lines \(A_iA_3\).

A characteristic element of a plane [straight-line] \(F\) of a submanifold \(\Psi_{\lambda-1}\) [5] of planes [straight lines] \((F)\) is a straight line [point] \(f\) such that the first differential neighborhood of it, for all submanifolds \(\Psi_1\) belonging to the submanifold \(\Psi_{\lambda-1}\), does not leave the plane [straight line] \(F\).

It follows from the expressions (14) and (15) that, under the fixing (13), the vertex \(A_4\) of the frame \(\{A_\alpha\}\) is placed at the point of intersection of the planes \(a_r\).

Here the case is excluded in which all the planes \(a_r\) belong to one pencil of planes. Under the fixing
\[ C_3^{ij}C_i^{36}-C_3^{i6}C_i^{3j}=1 \]
the case is excluded in which the tangent to the line
\[ (A_3)_{\omega_i=\omega_3=\omega_4=\omega_5=\omega_i^3=0} \]
passes through the point \(A_i\). The nonholonomic submanifold \(\Psi_5\) of conics
\[ \omega_\alpha^\beta=0 \]
is geometrically characterized by the fact that for it the point \(A_\alpha\) is

characteristic point of the plane \(A_\alpha A_\tau A_\mu\) \((\alpha,\beta,\tau,\mu=1,2,3,4;\ \alpha\ne\beta,\tau,\mu;\ \beta\ne\tau,\mu;\ \tau\ne\mu)\).

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

\[ \begin{gathered} dA_i=C_i^{i\lambda}\omega_\lambda A_i+\omega_\sigma A_j+C_i^{3\lambda}\omega_\lambda A_3+\omega_iA_4,\\ dA_3=C_3^{n\lambda}\omega_\lambda A_n+\omega_3A_4,\qquad dA_4=C_4^{\beta\lambda}\omega_\lambda A_\beta, \end{gathered} \tag{16} \]

where

\[ C_\beta^{\beta\lambda}=0,\qquad C_k^{k\lambda}-2C_3^{3\lambda}=\delta_6^\lambda . \]

The structure equations (2) lead to a system of exterior quadratic differential equations, whose compatibility follows directly from theorem (2) of [6]. The solution of this system depends on two functions of six arguments, which corresponds to the geometric fact that the family of conics \(K(3,6)\) exists with arbitrariness of two functions of six arguments.

From the systems of equations (9), (10) and formulas (16) the following propositions follow:

1) The nonholonomic congruence (submanifold \(\Psi_2\)) of conics \(\omega_i=\omega_3=\omega_5=\omega_6=0\) is geometrically characterized by the fact that, for it, the foci of the conic \(C\) coincide with the points of its intersection with the conic \(C_j^*\). For this congruence the point \(A_i\) is a triple focus of the conic \(C\) and a characteristic point of the line \(A_iA_3\). The remaining foci of the conic \(C\) correspond to the triple focal family \(\omega_j=0\). Moreover, the focal lines \(\omega_j=0\) are plane lines lying in the planes of the conics \(C\).

2) The nonholonomic congruence of conics \(\omega_i=\omega_3=\omega_4=\omega_6=0\) is geometrically characterized by the fact that, for it, the foci of the conic \(C\) are the points of its intersection with the conic \(C_i^*\) and the point \(A_i\). Moreover, the point \(A_i\) is a double focus, to which the focal family \(\omega_\sigma=0\) corresponds. The remaining foci correspond to the quadruple focal family \(\omega_j=0\). Moreover, the focal lines \(\omega_j=0\) are plane lines lying in the planes of the conics \(C\).

3) For the nonholonomic congruence of conics \(\omega_i=\omega_3=\omega_4=\omega_5=0\), the foci of the conic \(C\) coincide with the points \(A_k\). Moreover, the point \(A_i\) is a quadruple focus, and the point \(A_j\) a double focus, to which the focal family \(\omega_j=0\) corresponds. The tangent plane to the focal surface \((A_j)\) coincides with the plane \(A_jA_3A_4\), and the focal lines \(\omega_j=0\) on the surface \((A_j)\) are plane lines lying in the planes of the conics \(C\). For this congruence the conic point \(A_i\) is the characteristic point of the line \(A_iA_3\).

4) For the nonholonomic congruence of conics \(\omega_1=\omega_2=\omega_4=\omega_5=0\), the points \(A_i\) are triple foci of the conic \(C\) and characteristic points of the lines \(A_iA_3\).

5) On the nonholonomic surface \((A_3)_{\omega_j=\omega_3=\omega_4=\omega_5=0}\), the asymptotic lines coincide with the lines \(\omega_i\omega_3^i=0\). Moreover, the tangent to the line \(\omega_3^i=0\) coincides with the line \(A_3A_i\).

6) For the nonholonomic congruence of lines \((A_iA_3)_{\omega_i=\omega_3=\omega_4=\omega_5=0}\), the foci of the ray coincide with the points \(A_i\) and \(A_3\), and the torses are the ruled surfaces \(\omega_i\omega_3^j=0\). Moreover, the torses \(\omega_i=0\) degenerate into the planes of the conics \(C\), and the tangent planes to the focal surfaces \((A_i)\) coincide with the planes \(A_iA_3A_4\).

7) For the nonholonomic congruence of lines \((A_iA_3)_{\omega_j=\omega_3=\omega_x=\omega_i=0}\), one of the foci of the ray coincides with the point \(A_3\), and the ruled surfaces \(\omega_i\omega_3^{\,j}=0\) are torses. Moreover, the torses \(\omega_i=0\) degenerate into the planes of the conic \(C\).

8) For the nonholonomic congruence of lines \((A_iA_3)_{\omega_j=\omega_3=\omega_\sigma=\omega_i=0}\), the foci of the ray coincide with the points \(A_i\) and \(A_3\), and the torses coincide with the ruled surfaces \(\omega_i\omega_3^{\,j}=0\). Moreover, the torses \(\omega_i=0\) degenerate into the planes of the conic \(C\).

9) For the nonholonomic congruence of lines \((A_iA_4)_{\omega_i=\omega_u=\omega_5=\omega_z=0}\) \((u=j,3;\ z=x,6)\), one of the foci of the ray coincides with the point \(A_i\), and the torses are the ruled surfaces \(\omega_4^{\,3}\omega_4^{\,j}=0\). Moreover, the torses \(\omega_i^{\,3}=0\) degenerate into cones with vertices at \(A_i\). For this congruence the point \(A_i\) is a characteristic point of the line \(A_iA_3\).

Let us consider two classes of families \(K(3,6)\). The families of conics \(K_i(3,6)\), characterized by the natural equations \(C_3^{\,i6}=0\), exist and are determined up to an arbitrary function of six arguments. They have the following characteristic properties: the line \((A_3)_{\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=0}\) is a straight line coinciding with the line \(A_3A_j\); the line \((A_j)_{\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=0}\) degenerates into a point; the congruence of conics \(\omega_2=\omega_3=\omega_5=\omega_2^{\,3}=0\) is a holonomic congruence; the nonholonomic congruence of lines \((A_jA_3)_{\omega_i=\omega_3=\omega_4=\omega_5=0}\) degenerates into a ruled surface, whose tangent planes at the points \(A_j\) and \(A_3\) coincide with the planes \(A_jA_3A_4\) and \(A_1A_2A_3\), respectively.

REFERENCES

1. G. F. Laptev. Differential geometry of immersed manifolds. Proceedings of the Moscow Mathematical Society, 2, 1953, 275–382.

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

3. V. S. Malakhovskii. Nondegenerate congruences of curves of the second order in three-dimensional projective space. Geometrical Collection, issue 3 (Proceedings of Tomsk University, 168), 1963, 43–53.

4. V. I. Matveenko. Differential geometry of a six-parameter family of nondegenerate conics whose planes form a two-parameter family. Third Baltic Geometrical Conference. Abstracts of reports. Palanga, 1968, 114–115.

5. R. N. Shcherbakov. On the method of the moving frame and the framing of submanifolds. Geometrical Collection, issue 6 (Proceedings of Tomsk University, 191), 1967, 179–194.

6. V. V. Vasenin, R. N. Shcherbakov. On systems of exterior quadratic differential equations. Siberian Mathematical Journal.