FIBRATION OF \(\{2,3\}\)-COMPLEXES INTO CYCLIC SYSTEMS

M. R. VAINTRUB

(Presented by the Department of Higher Mathematics)

General complexes of circles in three-dimensional Euclidean space belong to a three-parameter family of planes. However, it is of interest to consider such complexes for which the planes of the circles form families of smaller dimension in comparison with the dimension of the manifold of circles itself. Such complexes of circles were considered in [1] and were called degenerate. In three-dimensional projective space, degenerate complexes of conics were studied in [2].

In the present paper, in three-dimensional Euclidean space we consider \(\{2,3\}\)-complexes (complexes of circles whose planes form a two-parameter family), which decompose into cyclic systems ([3], p. 107). Some properties of such complexes are established.

1. The closed system of differential equations of a \(\{2,3\}\)-complex, with each circle

\[ (x^1)^2+(x^2)^2-R^2=0,\qquad x^3=0 \tag{1} \]

of which a local canonical frame \((\bar A,\bar e_1,\bar e_2,\bar e_3)\), [1], is associated, can be written in the form

\[ \omega_1=\Gamma_{1i}\omega_3^i+\Gamma_{10}\Theta^0, \]

\[ \omega_2=\Gamma_{2i}\omega_3^i,\qquad \omega^3=\Gamma_{0i}^3\omega_3^i, \]

\[ \omega_1^2=\Gamma_0\Theta^0+\Gamma_i\omega_3^i, \tag{2} \]

\[ [\Delta\Gamma_{1i}\omega_3^i]+[\Delta\Gamma_{10}\Theta^0]=0, \]

\[ [\Delta\Gamma_{2i}\omega_3^i]=0,\qquad [\Delta\Gamma_{0i}^3\omega_3^i]=0, \]

\[ [\Delta\Gamma_0\Theta^0]+[\Delta\Gamma_i\omega_3^i]=0, \]

where \(\Gamma_{10}>0\) by virtue of the orientation of the vector \(\bar e_1\) of the frame, \(\Theta^0=-R\,dR\), \(\omega^i\equiv\omega_i\), and \((i,j,k=1,2;\ \text{do not sum over }k!)\)

\[ \Delta\Gamma_{0i}^3=d\Gamma_{0i}^3-\Gamma_{0k}^3\omega_i^k+\omega_i, \]

\[ \Delta\Gamma_{11}=d\Gamma_{11}-\Gamma_{12}\omega_1^2+\Gamma_1\omega^2-\omega^3,\qquad \Delta\Gamma_0=d\Gamma_0, \]

\[ \Delta\Gamma_{12}=d\Gamma_{12}-\Gamma_{11}\omega_2^1+\Gamma_2\omega^2,\qquad \Delta\Gamma_1=d\Gamma_1-\Gamma_2\omega_1^2, \]

\[ \Delta\Gamma_{10}=d\Gamma_{10}+\Gamma_0\omega^2,\qquad \Delta\Gamma_2=d\Gamma_2-\Gamma_1\omega_2^1+\omega_3^1, \tag{3} \]

\[ \Delta\Gamma_{21}=d\Gamma_{21}-\Gamma_{22}\omega_1^2-\Gamma_1\omega^1+\Gamma_0\Gamma_{11}\Theta^0, \]

\[ \Delta\Gamma_{22}=d\Gamma_{22}-\Gamma_{21}\omega_2^1-\Gamma_2\omega^1+\Gamma_0\Gamma_{12}\Theta^0-\omega^3. \]

It is assumed that the Pfaff forms \(\omega^\alpha,\ \omega_\alpha^\beta\) \((\alpha,\beta=1,2,3)\) satisfy the conditions of skew-symmetry and Cartan’s structure equations ([4], pp. 137–138).

The complex \(\{2.3\}\) is fibered into \(\infty^1\) congruences

\[ \omega \equiv a\omega_3^1+b\omega_3^2+c\Theta^0=0, \tag{4} \]

if equation (4) is completely integrable, that is,

\[ [D\omega,\omega]=0. \tag{5} \]

Putting

\[ \begin{aligned} da&=a_i\omega_3^i+a_3\Theta^0,\\ db&=b_i\omega_3^i+b_3\Theta^0,\\ dc&=c_i\omega_3^i+c_3\Theta^0, \end{aligned} \tag{6} \]

from (5) we obtain the condition

\[ (b_3-c_2+a\Gamma_0)a-(a_3-c_1-b\Gamma_0)b- \]

\[ -(b_1-a_2+a\Gamma_1+b\Gamma_2)c=0 \tag{7} \]

for the fibration of the complex \(\{2.3\}\) into congruences (4).

1. The circles of a holonomic congruence \(\omega=0\) of the complex \(\{2.3\}\) form a cyclic system if this congruence admits \(\infty^1\) surfaces orthogonal to all its circles. Arguing in the known way ([3], pp. 107–108), we obtain:

\[ \begin{aligned} R^2[\omega^3\omega_3^1]+[\Theta^0\omega^1]&\equiv 0\pmod\omega,\\ R^2[\omega^3\omega_3^2]+[\Theta^0\omega^2]&\equiv 0\pmod\omega,\\ R^2[\omega_3^1\omega_3^2]+[\omega^1\omega^2]&\equiv 0\pmod\omega. \end{aligned} \tag{8} \]

From (8), taking into account the system of differential equations (2), it follows that

\[ \begin{aligned} a\Gamma_{12}-b\Gamma_{11}+cR^2\Gamma_{02}^3&=0,\\ a\Gamma_{22}-b\Gamma_{21}-cR^2\Gamma_{01}^3&=0,\\ a\Gamma_{10}\Gamma_{22}-b\Gamma_{10}\Gamma_{21} -c\left(R^2+\Gamma_{11}\Gamma_{22}-\Gamma_{12}\Gamma_{21}\right)&=0. \end{aligned} \tag{9} \]

Analysis of equations (7) and (9) shows that an arbitrary complex \(\{2.3\}\) is not fibered into cyclic systems.

1. The following theorem holds. Theorem. If a complex of circles \(\{2.3\}\) is fibered into a one-parameter family of congruences \(\omega=0\), the circles of which form cyclic systems, then:

1) either one of the families of canal surfaces of this complex is a family of tubular surfaces;

2) or the radius of a circle of such a complex is the mean proportional between the distances from the center of the circle to the centers of its focal spheres.

Proof. A complex of circles \(\{2.3\}\) possesses two, in the general case distinct, families of canal surfaces [5]. The system of differential equations for finding the families of canal surfaces and the abscissas \(\rho\) of the centers of the focal spheres [6] of the complex \(\{2.3\}\) has the form:

\[ \begin{aligned} (\Gamma_{11}-\rho)\omega_3^1+\Gamma_{12}\omega_3^2+\Gamma_{10}\Theta^0&=0,\\ \Gamma_{21}\omega_3^1+(\Gamma_{22}-\rho)\omega_3^2&=0,\\ \rho(\Gamma_{01}^3\omega_3^1+\Gamma_{02}^3\omega_3^2)-\Theta^0&=0. \end{aligned} \tag{10} \]

The abscissas \(\rho\) of the centers of the focal spheres of this complex are determined from the equation

\[ (\Gamma_{10}\Gamma_{01}^{3}-1)\rho^{2} +\bigl[\Gamma_{11}+\Gamma_{22} +\Gamma_{10}(\Gamma_{02}^{3}\Gamma_{21}-\Gamma_{01}^{3}\Gamma_{22})\bigr]\rho +\Gamma_{12}\Gamma_{21}-\Gamma_{11}\Gamma_{22}=0. \tag{11} \]

If the condition of the formulated theorem is satisfied, then the system of equations (9) must have nonzero solutions. From the condition

\[ \det \begin{vmatrix} \Gamma_{12} & -\Gamma_{11} & R^{2}\Gamma_{02}^{3} \\ \Gamma_{22} & -\Gamma_{21} & -R^{2}\Gamma_{01}^{3} \\ -\Gamma_{10}\Gamma_{22} & \Gamma_{10}\Gamma_{21} & R^{2}+\Gamma_{11}\Gamma_{22}-\Gamma_{12}\Gamma_{21} \end{vmatrix} =0 \tag{12} \]

it follows that either

\[ \Gamma_{11}\Gamma_{22}-\Gamma_{12}\Gamma_{21}=0, \tag{13} \]

or

\[ R^{2}(1-\Gamma_{10}\Gamma_{01}^{3})+\Gamma_{11}\Gamma_{22}-\Gamma_{12}\Gamma_{21}=0. \tag{14} \]

Suppose condition (13) holds. Then it follows from (11) that one of the centers of the focal spheres of the complex coincides with the center of the circle of the complex. Substituting the value \(\rho=0\) into (10), we see that the family of canal surfaces enveloping this sphere is determined by the equations

\[ \Gamma_{11}\omega_{3}^{1}+\Gamma_{12}\omega_{3}^{2}=0,\qquad \Theta^{0}=0. \tag{15} \]

Since \(\Theta^{0}=-R\,dR\), (15) is a family of tubular surfaces.

Now suppose condition (14) is satisfied. Then equation (11) can be written in the form

\[ \rho^{2} +\frac{\Gamma_{11}+\Gamma_{22}+\Gamma_{10}(\Gamma_{02}^{3}\Gamma_{21}-\Gamma_{01}^{3}\Gamma_{22})} {\Gamma_{10}\Gamma_{01}^{3}-1}\,\rho -R^{2}=0. \tag{16} \]

It follows from (16) that \(|\rho_{1}|\,|\rho_{2}|=R^{2}\), that is, the second assertion of the theorem is valid.

We note that if conditions (13) and (14) are satisfied simultaneously, then, as follows from (11), for \(R\ne0\) there are degenerate ones among the canal surfaces of the complex \(\{2,3\}\).

REFERENCES

1. M. R. Vaintrub. Degenerate congruences and complexes of circles in three-dimensional Euclidean space. Deposited at VINITI, registration number 1231–69 Dep.

2. V. S. Malakhovskii. Complexes of curves of the second order in three-dimensional projective space. Proceedings of Tomsk University, vol. 176, Mechanics and Mathematics series, 1964, pp. 28–36.

3. S. P. Finikov. Theory of Congruences. Moscow–Leningrad, 1950.

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

5. M. R. Vaintrub. On canal surfaces of complexes of circles \((h,3)\). In: Materials for the Scientific Conference of Teachers of Mathematical Departments of Pedagogical Institutes of Siberia. Novokuznetsk, 1969, p. 80.

6. R. M. Geidelman. On the theory of a three-parameter complex of circles. Doklady of the Academy of Sciences of the USSR, no. 2, 1954, pp. 201–204.