Differential Geometry of the Family \(K(3,7)\) in \(P_3\)

V. I. Matveenko

(Presented by the Department of Higher Mathematics)

In the present article, in three-dimensional projective space, we consider the family \(K(3,7)\)—a seven-parameter family of nondegenerate conics whose planes form a three-parameter family. This family of conics exists with an arbitrary function of seven arguments. The notation and terminology correspond to those adopted in [1]—[5]. The indices take the following values:
\(i, j, k, g, t = 1, 2;\ i \ne j;\ k \ne t;\ m, n, r = 1, 2, 3;\ m \ne n;\ m \ne r;\ n \ne r;\ \alpha, \beta, \gamma = 1, 2, 3, 4;\ \lambda = 1, 2, 3, 4, 5, 6, 7;\ \sigma = i+3;\ x = j+3;\ \xi = 5+i;\ \eta = 5+j\). No summation is to be performed 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 \(C\), 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:

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

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

\[ \omega_n^4,\quad \omega_1^j,\quad \omega_3^j-\omega_i^3,\quad \omega_k^k-2\omega_3^3 \]

are the principal forms. We choose from them as basic forms

\[ \omega_n=\omega_n^4,\quad \omega_\sigma=\omega_i^j,\quad \omega_\xi=\omega_3^j-\omega_i^3. \]

The remaining principal form is expressed through them in the form

\[ \omega_k^k-2\omega_3^3=\Gamma^\lambda \omega_\lambda. \tag{4} \]

From this equation, by the usual method [1], one obtains the following system of differential equations of the intrinsic fundamental object \(\Gamma_1[\Gamma^\lambda]\):

\[ \delta \Gamma^l = \Gamma^l(\pi_4^4-\pi_i^i) - \Gamma^3\pi_3^l + \Gamma^\sigma\pi_4^3 - \Gamma^\xi\pi_4^i - \pi_4^l - \Gamma^l \sum_k \Gamma^{k+5}\pi_k^3, \]

\[ \begin{aligned} \delta\Gamma^\sigma &=\Gamma^\sigma(\pi_j^j-\pi_i^i)-\Gamma^\xi(\pi_3^i+\pi_j^3) -\Gamma^\sigma\sum_k \Gamma^{k+5}\pi_k^3,\\ \delta\Gamma^\xi &=\Gamma^\xi(\pi_j^j-\pi_3^3)-\Gamma^\sigma\pi_3^j -\Gamma^\xi\sum_k \Gamma^{k+5}\pi_k^3,\\ \delta\Gamma^3 &=\Gamma^3(\pi_4^4-\pi_3^3)+2\pi_4^3-\Gamma^k\pi_k^3 +\sum_k \Gamma^{k+5}(\pi_4^k-\Gamma^3\pi_k^3). \end{aligned} \tag{5} \]

This system of equations makes it possible to carry out the following normalization of the frame:
\[ \Gamma^4=\Gamma^5=0,\qquad \Gamma^6\ne0,\qquad \Gamma^7\ne0. \tag{6} \]

The forms \(\omega_3^j\) and \(\omega_i^j\) become principal forms, and their expansion in the basis forms may be written in the form
\[ \omega_3^j=\Gamma_3^{j\lambda}\omega_\lambda,\qquad \omega_i^j=\Gamma_3^{j\lambda}\omega_\lambda-\omega_\xi. \tag{7} \]

Let us clarify the geometric meaning of the normalization (6). The holonomic submanifold \(\Psi_4^*\) [5] of conics \(\omega_1=\omega_2=\omega_3=0\) is characterized geometrically by the fact that for it the plane of the conic \(C\) is fixed. From consideration of the system of equations
\[ (x^3)^2-2x^1x^2=0,\qquad x^4=0,\qquad x^n\omega_n=0, \]
\[ \Gamma^\lambda\omega_\lambda x^1x^2+ \sum_k\bigl[\omega_{k+3}(x^k)^2+\omega_{k+5}x^kx^3\bigr]=0 \tag{8} \]
for determining the points of intersection of the initial and adjacent conics of the family \(K(3,7)\), the following proposition follows: among the submanifolds \(\Psi_1^i\) [5] of conics
\[ \omega_1=\omega_2=\omega_3=0,\qquad \omega_4:\omega_5:\omega_6:\omega_7=b_1:b_2:b_3:b_4, \tag{9} \]
belonging to the submanifold \(\Psi_4^*\) of conics, there exist in the general case 480 such submanifolds \(\Psi_1^i\) of conics, for each of which the conic \(C\) has a fourfold characteristic point \(M_u\) \((u=1,2,3,\ldots,480)\). Under the normalization (6), the vertices \(A_i\) of the frame \(\{A_\alpha\}\) are placed at the points \(M_i\). The submanifold \(\Psi_1^i\) of conics \(\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=\omega_7=0\) is characterized geometrically by the fact that for it the point \(A_j\) is a fourfold characteristic point of the conic \(C\), whose plane is fixed for this submanifold. The submanifold \(\Psi_1^{i+2}\) of conics \(\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=\omega_\eta=0\) is characterized geometrically by the fact that for it the plane of the conic is fixed, while the points \(A_1\) and \(A_2\) are characteristic points, with the point \(A_j\) a double characteristic point of the conic \(C\). By the choice of the basis forms \(\omega_\lambda\), the cases are excluded in which the dimension of the manifold of planes of conics is less than three, in which the points \(A_i\) become double characteristic points of the conic \(C\) of the submanifolds \(\Psi_1^{i+2}\) of conics, and in which the characteristic points of the conic \(C\) for the submanifolds \(\Psi_1^i\) become indeterminate.

From the system of equations (7), in the usual way we obtain the following system of differential equations of the internally prolonged fundamental object \(\Gamma_2\{\Gamma_1,\Gamma_3^{i\lambda}\}\) [1]:
\[ \begin{aligned} \delta\Gamma_3^{gi} &=\Gamma_3^{gi}(\pi_3^3+\pi_4^4-\pi_g^g-\pi_i^i) +\Gamma_3^{g\sigma}\pi_4^j-\Gamma_3^{g\xi}\pi_4^3,\\ \delta\Gamma_3^{i3} &=\Gamma_3^{i3}(\pi_4^4-\pi_i^i)+(\Gamma_3^{i\eta}-1)\pi_4^i+\Gamma_3^{i\xi}\pi_4^i,\\ \delta\Gamma_3^{i\sigma} &=\Gamma_3^{i\sigma}(\pi_j^j-2\pi_i^i+\pi_3^3),\qquad \delta\Gamma_3^{i\xi}=\Gamma_3^{i\xi}(\pi_j^j-\pi_i^i),\\ \delta\Gamma_3^{ix} &=\Gamma_3^{ix}(\pi_3^3-\pi_j^j),\qquad \delta\Gamma_3^{i\eta}=0. \end{aligned} \tag{10} \]

Since this system of equations is algebraically solvable with respect to all independent secondary forms \(\pi_\alpha^\beta=\pi_\alpha^\beta-\delta_\alpha^\beta\pi_4^4\), the object \(\Gamma_2\) is fundamental [1]. From the systems of equations (5) and (10) we note directly that the quantities \(\Gamma^\xi,\ \Gamma^{k\sigma},\ \Gamma_3^{j\xi}\) are relative invariants. The condition \(\Gamma^\xi=0\) characterizes the family of conics for which the point \(A_j\) is a triple characteristic point of the conic \(C\) of the submanifold \(\Psi_1^{i+2}\). The condition \(\Gamma_3^{k\sigma}=0\) \([\Gamma_3^{j\xi}=0]\) characterizes the family of conics for which the tangent \(l^\sigma\) \([l^\xi]\) to the line
\((A_3)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=\omega_7=0}\)
\([(A_3)_{\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=\omega_\eta=0}]\) coincides with the line \(A_3A_i\) \([A_3A_j]\).

The quantities

\[ \Gamma_3^{j\xi},\quad E=(\Gamma_3^{j\sigma}-1)\Gamma_3^{j\chi},\quad L_i=\frac{\Gamma_3^{i\xi}}{(\Gamma^\xi)^2\Gamma_3^{j\xi}},\quad H_i=\Gamma^1\Gamma_3^{j\sigma}. \]

are absolute invariants. Geometrically they are characterized through cross-ratios of quadruples of points as follows:

\[ E=-(A_jA_3;\,B_i^\sigma B^j),\qquad \Gamma_3^{j\xi}=\frac{1}{1+(A_iA_j;\,B_\xi B_3^\xi)}, \]

\[ H_i=(A_iA_3;\,M_{jj}B^i),\qquad L_i=\frac{1}{2}(A_iA_j;\,M_{i3}B_3^\sigma), \tag{11} \]

where \(B_3^\sigma\) \([B_3^\xi]\) is the point of intersection of the lines \(A_1A_2\) and \(l^\sigma\) \([l^\xi]\), \(B_i^\sigma\) is the point of intersection of the line \(A_jA_3\) with the tangent \(l_i\) to the line
\((A_i)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=\omega_7=0}\),
\(B^i\) is the point of the envelope of the family of lines
\((A_iA_3)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=\omega_7=0}\),
\(M_{ir}\) is the projection from the point \(A_r\) onto the line \(A_mA_n\) of the characteristic point
\(M^i=(\Gamma^\xi)^2A_i+2A_j-2\Gamma^\xi A_3\) of the conic \(C\) of the submanifold \(\Psi_1^{i+2}\) of conics. Using expressions (11), we obtain the following equalities:

\[ (A_1A_3B_2^5B^1)=(A_2A_3;\,B_1^4B^2),\qquad (A_1A_2;\,B_3^\sigma B_\sigma)=-1, \]

\[ (A_1A_3M_{12}B^1)\ A_2A_3;\,M_{11}B^2) =-2(A_1A_3;\,B_2^5B^1), \]

\[ (A_iA_3;\,M_{jj}B^i)=-2(A_jA_3;\,B_i^\sigma M_{ii}), \]

where \(B_\sigma\) \([B_\xi]\) is the point of the envelope of the family of lines

\[ (A_1A_2)_{\omega_1=\omega_2=\omega_3=\omega_x=\omega_6=\omega_7=0} \quad \bigl[(A_1A_2)_{\omega_1=\omega_2=\omega_3=\omega_4=\omega_5=\omega_\eta=0}\bigr]. \]

The system of equations (10) makes it possible to carry out the following fixation of the frame:

\[ \Gamma_3^{i3}=1,\qquad \Gamma_3^{i3}\Delta+\sum_k \Gamma_3^{i,k+5}\Delta_k=0, \]

\[ \sum_k\bigl[a\Gamma_3^{ik}+a_{k+5}^k(\Gamma_3^{i,k+5}-1)+a_{k+5}^i\Gamma_3^{k,k+5}\bigr]=0, \tag{12} \]

where

\[ a=\Gamma_3^{16}\Gamma_3^{27}-\Gamma_3^{26}\Gamma_3^{17}\ne0,\qquad a_\xi^k=\Gamma_3^{jk}\Gamma_3^{i\eta}-\Gamma_3^{ik}\Gamma_3^{j\eta}, \]

\[ \Delta=(\Gamma_3^{26}-1)(\Gamma_3^{17}-1)-\Gamma_3^{16}\Gamma_3^{27}\ne0,\qquad \Delta_i=\Gamma_3^{i3}\Gamma_3^{j\eta}-\Gamma_3^{j3}(\Gamma_3^{i\eta}-1). \]

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. We denote by \(l_3\) the tangent to the line

\[ (A_3)_{\omega_1=\omega_2=\omega_4=\omega_5=\omega_1^3=\omega_2^3=0}, \]

and by \(b_i\) the tangent plane to the holonomic

surfaces \((A_i)_{\omega_j=\omega_3=\omega_\lambda=\omega_3^1=\omega_3^2=0}\), and by \(R_i\) the point of intersection of the line \(l_3\) with the plane \(b_i\). The submanifold \(\Psi_1\) of the conics \(\omega_1=\omega_2=\omega_4=\omega_5=\omega_3^1=\omega_3^2=0\) is characterized geometrically by the fact that, for it, the points \(A_i\) are fixed. The holonomic submanifold \(\Psi_2\) of the conics \(\omega_j=\omega_3^1=\omega_\lambda=\omega_3^1=\omega_3^2=0\) is characterized geometrically by the fact that, for it, the point \(A_3\) and the line \(A_jA_3\) are fixed. Under the fixation (12), the point \(A_4\) is chosen on the line \(l_3\) so that \((R_1R_2;\,A_3A_4)=-1\). If the points \(R_1\) and \(R_2\) coincide, then the point \(A_4\) coincides with them. Under the fixation \(\Gamma_3^{13}=0\), the case is excluded in which the point \(Q=\Gamma_3^{13}A_1-\Gamma_3^{23}A_2\) of the edge of regression of the torsus \((A_1A_2)_{\omega_1=\omega_2=\omega_4=\omega_5=\omega_6=\omega_7=0}\) coincides with the points \(A_i\). The inequality \(a\ne0\) excludes the case of coincidence of the lines \(l^6\) and \(l^7\).

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

\[ dA_i=\Gamma_i^{i\lambda}\omega_\lambda A_i+\omega_\sigma A_j+ (\Gamma_3^{j\lambda}\omega_\lambda-\omega_\xi)A_3+\omega_iA_4, \tag{13} \]

\[ dA_3=\Gamma_3^{n\lambda}\omega_\lambda A_n+\omega_3A_4,\qquad dA_4=\Gamma_4^{\beta\lambda}\omega_\lambda A_\beta \]

where

\[ \Gamma_\beta^{\beta\lambda}=0,\qquad (\Gamma_k^{k6}-2\Gamma_3^{36})\Gamma_3^{15} = (\Gamma_k^{k7}-2\Gamma_3^{37})\Gamma_3^{24}. \]

Using the constructed canonical frame \(\{A_\alpha\}\), one may consider various submanifolds of the family \(K(3,7)\). Consider, for example, the submanifold \(\Psi_2\) (a holonomic congruence) of the conics \(\omega_j=\omega_3^1=\omega_\lambda=\omega_3^1=\omega_3^2=0\). For this congruence the point \(A_j\) is a triple focus of the conic \(C\). The remaining foci of the conic \(C\) correspond to the triply focal family \(\omega_i=0\). Moreover, the focal lines \(\omega_i=0\) are plane lines lying in the planes of the conics \(C\). For this congruence of conics the point \(A_j\) is a characteristic point of the line \(A_jA_3\). The nonholonomic submanifold \(\Psi_6'\) [5] of the conics \(\omega_\alpha^\rho=0\) is characterized geometrically by the fact that, for it, the point \(A_\alpha\) is a characteristic point of the plane \(A_\alpha A_\nu A_\mu\) \((\alpha,\rho,\nu,\mu=1,2,3,4;\ \alpha\ne\rho,\nu,\mu;\ \rho\ne\nu,\mu;\ \nu\ne\mu)\).

References

1. G. F. Laptev. Differential geometry of immersed manifolds. Transactions 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. Geometricheskii sbornik, issue 3 (Transactions 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 Geometric Conference. Abstracts of Reports. Palanga, 1968, 114–115.

5. R. N. Shcherbakov. On the method of the moving frame and reframing of submanifolds. Geometricheskii sbornik, issue 6 (Transactions of Tomsk University, 191), 1967, 179–194.