On Nonholonomic Gambier Congruences
Abstract Generated abstract
This paper studies nonholonomic congruences of circles in three-dimensional Euclidean space, focusing on complexes whose circle planes form a three-parameter family and on the case in which cyclic points are double foci. Using Cartan’s moving frame method and Pfaffian systems, it formulates the differential structure of such circle complexes, defines principal diameters and principal points, and derives equations for the foci of a general nonholonomic circle congruence. The analysis characterizes nonholonomic Gambier congruences by explicit invariant conditions and relates them to associated nonholonomic congruences of axes. It proves that a nonholonomic congruence is Gambier precisely when the boundary points of the corresponding axis congruence coincide, and that such a congruence corresponds to a bicentral nonholonomic congruence in the axis complex.
Full Text
On Nonholonomic Gambier Congruences
M. R. Vaintrub
(Presented by the Department of Higher Mathematics)
The paper studies nonholonomic congruences of circles of a complex, in which the cyclic points are double foci. A connection is established between such congruences and nonholonomic line congruences associated with a complex of axes.
- In three-dimensional Euclidean space \(E_3\), complexes of circles are considered whose planes form a three-parameter family (complexes \(\{3.3\}\)). With each circle of a complex \(\{3.3\}\) there is associated a local orthonormal frame \((\bar A, e_1, e_2, e_3)\), whose origin \(\bar A\) is placed at the center of the current circle of the complex, while the vectors \(e_1\) and \(e_2\) lie in the plane of this circle. The infinitesimal displacement of the frame is determined by the equations
\[ d\bar A=\omega^\alpha e_\alpha,\qquad de_\alpha=\omega_\alpha^\beta e_\beta,\qquad \alpha,\beta,\gamma=1,2,3, \tag{1} \]
where the Pfaffian forms \(\omega^\alpha,\omega_\alpha^\beta\) satisfy the skew-symmetry conditions and Cartan’s structure equations ([1], pp. 137–138). With respect to the frame chosen in this way, the equations of the current circle of the complex \(\{3.3\}\) are written in the form
\[ (x^1)^2+(x^2)^2-R^2=0,\qquad x^3=0. \tag{2} \]
The six Pfaffian forms \(\omega^i=\omega_i,\ \omega_3^i,\ \omega^3,\ \Theta_0=-R\,dR,\ (i,j=1,2)\) are the primary forms of the circle in \(E_3\). Let us choose as basic forms \(\omega_3^i,\omega^3\), which determine the position of the plane of the circle in space. Then the closed system of differential equations of the complex \(\{3.3\}\) will be written in the form:
\[ \Theta_0=\Gamma_{0i}\omega_3^i+\Gamma_{03}\omega^3, \]
\[ \omega_i=\Gamma_{ij}\omega_3^j+\Gamma_{i3}\omega^3, \]
\[ [\Delta\Gamma_{0i}\omega_3^i]+[\Delta\Gamma_{03}\omega^3]=0, \tag{3} \]
\[ [\Delta\Gamma_{ij}\omega_3^j]+[\Delta\Gamma_{i3}\omega^3]=0, \]
where
\[ \Delta\Gamma_{0i}=d\Gamma_{0i}-\Gamma_{0k}\omega_i^k-\Gamma_{03}\omega_i, \]
\[ \Delta\Gamma_{03}=d\Gamma_{03}, \]
\[ \Delta\Gamma_{ii}=d\Gamma_{ii}-(\Gamma_{ik}+\Gamma_{ki})\omega_i^k-\Gamma_{i3}\omega_i, \tag{4} \]
\[ \Delta\Gamma_{ik}=d\Gamma_{ik}+(\Gamma_{ii}-\Gamma_{kk})\omega_i^k-\Gamma_{i3}\omega_k, \]
\[ \Delta\Gamma_{i3}=d\Gamma_{i3}-\Gamma_{k3}\omega_i^k-\omega_i^3, \]
\[ (k\ne i=1,2;\ \text{do not sum over }k!) \]
Carrying out the construction of a chain of integral elements with respect to the basis forms \(\omega^1_3,\ \omega^2_3,\ \omega^3\), we are convinced that the complexes \(\{3.3\}\) exist and are determined with arbitrariness of 3 functions of 3 arguments.
2. Definition 1. The principal diameter of a circle of the complex \(\{3.3\}\) is a diameter to which there corresponds a one-parameter family of circles with a tangent to the line of centers orthogonal to this diameter.
Equations of the form
\[ \Gamma_{12}(x^1)^2+(\Gamma_{22}-\Gamma_{11})x^1x^2-\Gamma_{21}(x^2)^2=0,\quad x^3=0 \tag{5} \]
determine two real, two imaginary, or one principal diameter.
Definition 2. The points of intersection of a circle with its principal diameters are called the principal points of a circle of the complex. Using the principal diameters and the principal points of the circle, one can carry out the final canonization of the frame of the complex \(\{3.3\}\), directing the vector \(\bar e_1\) of the frame either to a principal point or along the bisector of the angle between the principal directions.
One not completely integrable Pfaff equation
\[ \omega^3=0 \]
may be regarded as the equation of a nonholonomic surface orthogonal to the axis of the circle and described by its center. Seeking the lines of curvature of the second kind on the nonholonomic surface of centers \((A)_{\omega^3=0}\), we are convinced that to them there correspond submanifolds associated with the principal diameters of the circle of the complex. This makes it possible to supplement the geometric picture of the canonization of the frame. In what follows we shall assume that with each circle of the complex \(\{3.3\}\) there is associated a certain canonical frame.
3. An arbitrary nonholonomic congruence of circles of the complex \(\{3.3\}\) is determined by one not completely integrable Pfaff equation [2].
\[ \omega \equiv a\omega^1_3+b\omega^2_3-c\omega^3=0, \tag{6} \]
where
\[ a,\ b,\ c \]
are invariants of the nonholonomic congruence (6), which are functions of the primary parameters.
Seeking the foci of the nonholonomic congruence \((\bar A e_3)_{\omega=0}\) corresponding to the congruence (6), we are convinced that to congruences with \(c=0\) there correspond nonholonomic cylindrical congruences in the complex of axes. Excluding such congruences from consideration, we shall set \(c=1\).
The foci and focal families of the congruence \(\omega=0\) are determined by the system of equations:
\[ (x^1)^2+(x^2)^2-R^2=0,\quad x^3=0, \]
\[ (x^i\Gamma_{i1}-\Gamma_{01})\omega^1_3+(x^i\Gamma_{i2}-\Gamma_{02})\omega^2_3+(x^i\Gamma_{i3}-\Gamma_{03})\omega^3=0, \]
\[ x^1\omega^1_3+x^2\omega^2_3-\omega^3=0, \]
\[ a\omega^1_3+b\omega^2_3-\omega^3=0. \tag{7} \]
In what follows we shall assume that
\[ A=b\Gamma_{13}+\Gamma_{12},\quad B=a\Gamma_{23}+\Gamma_{21}, \]
\[ C=b\Gamma_{23}-a\Gamma_{13}+\Gamma_{22}-\Gamma_{11}, \]
\[ D=b(\Gamma_{11}-\Gamma_{03})-a\Gamma_{12}-\Gamma_{02}, \]
\[ E=b\Gamma_{21}-a(\Gamma_{22}-\Gamma_{03})+\Gamma_{01}, \]
\[ F=a\Gamma_{02}-b\Gamma_{01}. \tag{8} \]
Eliminating from equations (7) the ratio \(\omega_3^1:\omega_3^2:\omega^3\) of the basis forms, we obtain the system of equations
\[
(x^1)^2+(x^2)^2-R^2=0,\qquad x^3=0,
\]
\[
A(x^1)^2-B(x^2)^2+Cx^1x^2+Dx^1+Ex^2+F=0,
\tag{9}
\]
which determines the coordinates of the foci of the congruence. This system is equivalent to the equations
\[
(x^1)^2+(x^2)^2-R^2=0,\qquad x^3=0,
\]
\[
[C^2+(A+B)^2](x^2)^4+[2CD-2E(A+B)](x^2)^3+
\]
\[
+[D^2-C^2R^2+E^2-2(A+B)(F+AR^2)](x^2)^2+
\tag{10}
\]
\[
+[2E(F+AR^2)-2CDR^2]x^2+(F+AR^2)^2-R^2D^2=0.
\]
4. Definition 3. Congruences of Gambier [3, 4], nonholonomic or holonomic, are those congruences of circles whose cyclic points are double foci. From (8) and (10) it follows that such congruences are characterized by the conditions
\[ A+B=0,\quad C=0 \tag{11} \]
and exist in an arbitrary complex.
Theorem 1. A nonholonomic congruence of the complex \(\{3.3\}\) is a Gambier congruence if and only if the boundary points of the corresponding nonholonomic congruence of axes coincide.
Proof. Arguing in the usual way ([5], p. 52), to find the abscissas of the boundary points of the nonholonomic congruence \((\overline{A}e_3)_{\omega=0}\), we obtain a quadratic equation whose discriminant is determined by the formula
\[ \overline{D}=C^2+(A+B)^2. \tag{12} \]
The validity of the assertion of the theorem follows from comparing conditions (11) with the conditions for the discriminant (12) to vanish.
Theorem 2. To a nonholonomic Gambier congruence belonging to the complex \(\{3.3\}\) there corresponds a bicentral [6] nonholonomic congruence in the complex of axes.
Proof. A bicentral nonholonomic congruence in the complex of axes is determined by the condition
\[
2\Gamma_{13}\Gamma_{23}(\Gamma_{12}+\Gamma_{21})+(\Gamma_{11}-\Gamma_{22})[(\Gamma_{13})^2-(\Gamma_{23})^2]+
\]
\[
+(a\Gamma_{13}+b\Gamma_{23})[(\Gamma_{13})^2+(\Gamma_{23})^2]=0.
\tag{13}
\]
Using (8), from (11) we obtain:
\[
a=-\frac{\Gamma_{13}(\Gamma_{22}-\Gamma_{11})-\Gamma_{23}(\Gamma_{12}+\Gamma_{21})}
{(\Gamma_{13})^2+(\Gamma_{23})^2},
\]
\[
b=-\frac{\Gamma_{13}(\Gamma_{12}+\Gamma_{21})+\Gamma_{23}(\Gamma_{22}-\Gamma_{11})}
{(\Gamma_{13})^2+(\Gamma_{23})^2}.
\tag{14}
\]
Substituting the values of \(a\) and \(b\) from (14) into condition (13), we verify the validity of the assertion of the theorem.
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