DIFFERENTIAL GEOMETRY OF A FIVE-PARAMETER FAMILY OF NONDEGENERATE CONICS WHOSE PLANES FORM A ONE-PARAMETER FAMILY

V. I. MATVEEVKO

(Presented by the Department of Geometry of Tomsk University)

In the present article, in three-dimensional projective space we consider a manifold \(K\) \((1,5)\)—a five-parameter family of nondegenerate conics whose planes form a one-parameter family. This family of conics exists with an arbitrary function of five arguments. The notation and terminology correspond to those adopted in [1]—[4].

Place the vertices \(A_i\) \((i=1,2)\) of the coordinate tetrahedron \(\{A_\alpha\}\) \((\alpha=1,2,3,4)\) at the points of intersection of the conic \(C\) with the characteristic of its plane, the vertex \(A_3\) at the point of intersection of the tangents to the conic at the points \(A_i\), and the vertex \(A_4\) at an arbitrary point of the projective space not lying in the plane of the conic. In this case the case of tangency of the conic \(C\) with the characteristic of its plane is excluded. The derivation formulas of the frame \(\{A_\alpha\}\) have the form:

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

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

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

Here and below we agree to assume that \(i,j,\kappa,t=1,2;\ i\ne j;\ \kappa\ne t;\ m,n,r=1,2,3;\ m\ne n;\ m\ne r;\ n\ne r;\ \alpha,\beta,\gamma=1,2,3,4;\ \lambda=1,2,3,4,5\). There is no summation over \(i,j,\beta\)! The equations of the conic \(C\) with respect to this frame have the form

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

The forms

\[ \omega_i^j,\ \omega_i^3,\ \omega_n^4,\ \omega_3^i,\ \omega_\kappa^\kappa-2\omega_3^3-d\ln p \]

are principal. We take the following of them as basic forms:

\[ \omega_i=\omega_i^j,\quad \omega_0=\omega_3^j-\frac{1}{p}\omega_i^3,\quad \omega_5=\omega_3^4. \]

Here and below the notation \(\varepsilon=i+2,\ \chi=j+2\) is used. The remaining principal forms are expressed through the basic ones in the form

\[ \omega_i^4=0,\quad \omega_i^3=C_i^{35}\omega_5,\quad \omega_3^j=\omega_\chi+C_j^{35}\omega_5, \]

\[ \omega_\kappa^\kappa-2\omega_3^3-d\ln p=a^\lambda\omega_\lambda. \tag{4} \]

A characteristic conic \(C_q^*\) [4] is a conic that passes through the point \(A_3\) and the characteristic points of the conic \(C\) for \(\omega_h=\omega_g=\omega_l=\omega_s=0\). Here and below we shall agree to assume that \(g,h,l,q=1,2,3,4;\ g\ne h,l,q;\ h\ne l,q;\ l\ne q\). The one-parameter family of conics \(\omega_i=\omega_3=\omega_4=\omega_5=0\) is characterized geometrically by the fact that, for it, the points \(A_3\) and \(A_i\) are fixed. The one-parameter family of conics \(\omega_1=\omega_2=\omega_5=\omega_5=0\) is characterized geometrically by the fact that, for it, the points \(A_1,A_2\) and the line \(A_3A_i\) are fixed. When the \(\omega_\lambda\) are chosen as basis forms, the cases in which the characteristic conics \(C_i^*\) pass through the points \(A_i\), and the cases of polar conjugacy of the points \(A_i\) and \(A_3\) with respect to the characteristic conics \(C_\sigma^*\), are excluded. Differentiating the equations (4) exteriorly, in the usual way we obtain the following system of differential equations of the inner local object \(C_1[p,a^\lambda,C_i^{35}]\) [1]:

\[ \delta\ln p=\pi_k^k-2\pi_3^3,\qquad \delta a^i=a^i(\pi_j^j-\pi_i^i), \]

\[ \delta a^s=a^s(\pi_j^j-\pi_3^3),\qquad \delta C_i^{35}=C_i^{35}(\pi_i^i+\pi_4^4-2\pi_3^3), \tag{5} \]

\[ \delta a^5=(\pi_k^k-2\pi_3^3)(a^3C_1^{35}+a^4C_2^{35})+a^3\pi_4^2+a^4\pi_4^1+a^5(\pi_3^3-\pi_4^4)+2\pi_4^3 . \]

From these equations we directly conclude that the quantities \(p,a^q,C_i^{35}\) are relative invariants. The condition \(C_i^{35}=0\) characterizes families of conics for which \(M\equiv A_i\), where \(M\) is the cusp point of the envelope edge of the plane of the conics of the torsal. The condition \(a^i=0\) characterizes families of conics for which the characteristic conic \(C_i^*\) degenerates into the double line \(A_jA_3\). The condition \(a^s=0\) characterizes families of conics for which the characteristic conic \(C_s^*\) degenerates into the pair of lines \(A_jA_3\) and \(A_1A_2\).

The quantities

\[ B_i=\frac{a^iC_i^{35}}{C_j^{35}},\qquad E_i=\frac{C_i^{35}(a^s)^2}{pC_j^{35}} \]

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

\[ B_i=(A_iA_j;MS_{i3}),\qquad E_i=-2(A_iA_j;MS_{s3}), \tag{6} \]

where \(S_{hn}\) is the projection from the point \(A_n\) onto the line \(A_mA_r\) of one of the points of intersection of the conics \(C\) and \(C_h^*\). From (6) there follow the following equalities:

\[ (A_1A_3;S_{42}S_{32})=(A_2A_3;S_{31}S_{41}), \]

\[ (A_1A_3;S_{22}S_{12})=(A_2A_3;S_{11}S_{21}), \]

\[ (A_1A_2;S_{23}S_{13})=(A_iA_3;S_{jj}S_{ij})^2, \]

\[ (A_1A_2;MS_{33})(A_2A_1;MS_{43})=(A_iA_3;S_{sj}S_{aj})^2, \]

which are valid for a general family of conics \(K(1,5)\).

Excluding from consideration the cases in which the relative invariants \(C_1^{35}\) and \(a^1\) vanish, we carry out the following fixation of the frame:

\[ C_1^{35}=p=-a^1=1. \]

If we also take into account the condition of equiprojectivity \((A_1A_2A_3A_4)=1\), then we obtain \(\pi_3^3=0\). The forms \(\omega_\beta^3\) become principal, and their expansion in the basis forms may be written in the form \(\omega_\beta^3=C_\beta^{3\lambda}\omega_\lambda\), where \(C_\alpha^{3\lambda}=0\).

Place the vertex \(A_4\) of the frame \(\{A_\alpha\}\) at the point of intersection of the planes \(a_n\), where \(a_i\) are the tangent planes to the nonholonomic surfaces-

...planes that describe the points \(S_{1i}\), and \(a_3\) is the tangent plane to the developable surface described by the point \(A_3\) under

\[ \omega_1=\omega_2=C^\lambda \omega_\lambda=0 \qquad (C^\lambda=C_1^\lambda-C_2^\lambda). \tag{7} \]

Here the case is excluded in which all planes \(a_n\) belong to a single pencil of planes. The nonholonomic congruence of conics (7) is characterized geometrically by the fact that, for it, the points \(A_i\) and \(S_{13}\) are the points of the envelopes of the families of lines \((A_iA_3)\) and \((S_{13}A_3)\), respectively. The derivation formulas (1) of the canonical frame thus obtained have the form:

\[ dA_i=C_i^{i\lambda}\omega_\lambda A_i+\omega_i A_j+C_i^{35}\omega_5 A_3, \qquad dA_4=C_4^{\alpha\lambda}\omega_\lambda A_\alpha, \]

\[ dA_3=\sum_{\kappa}(\omega_{\kappa+2}+C_\kappa^{35}\omega_5)A_\kappa +C_3^{3\lambda}\omega_\lambda A_3+\omega_5 A_4, \]

where

\[ C_\kappa^{\kappa 1}-2C_3^{21}=-1, \qquad C_2^{2\alpha}+3C_3^{3\alpha}=\delta_1^\alpha C_2^{35}, \]

\[ C^4\!\left[\sqrt{2}\,C_1^{13}C_2^{35} +C^3(2C_2^{35}-1-\sqrt{2}\,C_3^{35}) +\sqrt{2}\,C_3^{33}(C^5+3C_1^{15})\right] \]

\[ -(C^5-C^3)\!\left[C_3^{33}(C^4+3C_1^{14}) -C_3^{34}(C^3+3C_1^{13})+\sqrt{2}\,C^3\right]=0, \]

\[ C^3\!\left(\sqrt{2}\,C_2^{35}C_3^{34} -\sqrt{2}\,C_3^{25}-C_2^{35}\right) +\sqrt{2}\,C_3^{33}(C^5-C^4C_2^{35})+ \]

\[ +3C_3^{33}C_1^{15}+C_2^{25}C_1^{13} +3C_2^{35}(C_1^{13}-C_1^{14})=0, \]

\[ C_2^{35}C^4-C^5+C^3=0 \qquad (C^\lambda=C_1^\lambda-C_2^\lambda,\quad C^3\ne0,\quad C_2^{35}\ne1). \]

REFERENCES

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

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