R. N. Shcherbakov

PROJECTIVELY INVARIANT FRAMES OF A RULED SURFACE BELONGING TO A GIVEN CONGRUENCE

(Presented by Academician P. S. Aleksandrov, 3 IX 1956)

In the present paper we consider projectively invariant frames of a ruled surface belonging to a given congruence (analogous constructions in metric and affine geometries were carried out in ((^1,\ ^2))). The notation, for the most part, corresponds to that adopted in ((^3,\ ^4)).

§ 1. Construction of frames. Let (A_1A_2) be a ray of the congruence, and let (A_1, A_2, A_3, A_4) be the vertices of a frame. In the derivation formulas (dA_i=\omega_i^k A_k), the Pfaff forms (\omega_i^k=a_i^k\omega_1^3+b_i^k\omega_1^4) satisfy the well-known structure equations (D\omega_i^k=[\omega_i^j\omega_j^k]). Taking (\omega_1^3) and (\omega_1^4) as the fundamental forms and carrying out the first step of fixing the frame, we obtain: (\omega_2^3=\omega_1^4), (\omega_2^4=\omega_1^3). In this case the foci (F_i) of the ray are determined by the equalities (F_1=A_1+A_2), (F_2=A_1-A_2), and the torses of the congruence by the equation ((\omega_1^3)^2-(\omega_1^4)^2=0).

Two ruled surfaces (S_1) and (S_2), passing through the given ray of the congruence and conjugate in the sense of Sannia ((^5)), can be characterized in projective-differential geometry by the following property: the tangent plane of the surface (S_1) at any point (M_1) of the ray is the tangent plane of the surface (S_2) at the point (M_2) of the ray harmonically conjugate to (M_1) with respect to the foci of the ray; moreover, the tangent plane of the surface (S_1) at (M_2) is the tangent plane of the surface (S_2) at (M_1).

With our fixation the equation (\omega_1^3\omega_1^4=0) singles out the conjugate net of ruled surfaces belonging to the congruence, while the planes (A_1A_2A_3) and (A_1A_2A_4) are the tangent planes of these surfaces at the points (A_1) and (A_2).

At the second step of fixing, two variants arise. In both variants the points (G_1=A_3+A_4) and (G_2=A_3-A_4) fall on the rays of the congruences (K_1) and (K_2), which are the Laplace transforms of the given congruence by means, respectively, of the first and second focal nets; and a normalization is carried out that excludes from consideration ruled surfaces for which at least one focal line degenerates.

In the first variant (the (K)-frame) the fixation is performed so that the lines (A_1A_4) and (A_2A_3) belong to the osculating complex of the net for the given ray (i.e. to the complex of lines determined by the given ray and by two pairs of infinitely close rays of ruled surfaces of the net passing through this ray), while in the second (the (M)-frame) it is performed so that the vertices are the median ((^6)) points of the ray with respect to the foci.

The last step of the fixation is carried out so that the line (A_3A_4) becomes the line joining the second foci of the rays of the congruences (K_1) and (K_2) (these foci coincide with the points (G_1) and (G_2)). Adding also the usual normalization ((A_1A_2A_3A_4)=1), we obtain the following relations for the coefficients (a_i^k) and (b_i^k):

[
b_1^3=a_1^4=a_2^3=b_2^4=0,\qquad
a_1^3=b_1^4=b_2^3=a_2^4=1,
]

[
a_1^1+a_2^2+a_3^3+a_4^4=b_1^1+b_2^2+b_3^3+b_4^4=0,
]

[
\begin{gathered}
a_1^2-a_2^1=b_1^1-b_2^2=A,\qquad
b_1^2-b_2^1=a_1^1-a_2^2=B,\
a_4^3-a_3^4=b_3^3-b_4^4=A^,\qquad
b_4^3-b_3^4=a_3^3-a_4^4=B^,\
a_3^2-a_4^1=b_1^2-b_3^1,\qquad
b_3^2-b_4^1=a_4^2-a_3^1,\
B^2-A^2=c=\mathrm{const},\qquad A+\varepsilon A^*=0,
\end{gathered}
]

where (\varepsilon=+1) for the (K)-frame and (\varepsilon=-1) for the (M)-frame.

Taking these relations into account, the integrability conditions of the derivational formulas will contain 11 exterior differential equations in 14 functions and will determine a congruence referred to an arbitrary net of conjugate ruled surfaces, with an arbitrariness of 3 functions of 2 arguments.

Putting (\omega_1^4=0,\;(\omega_1^3){\omega_1^4=0}=ds,\;(a_i^k)=\alpha_i^k), we obtain 2 frames of a ruled surface belonging to the given congruence. The derivational formulas of these frames will have the form:

[
dA_i/ds=\alpha_i^k A_k,
\tag{1}
]

[
\alpha_1^4=\alpha_2^3=0,\qquad
\alpha_1^3=\alpha_1^4=1,\qquad
(\alpha_1^1-\alpha_2^2)^2-(\alpha_1^2-\alpha_2^1)^2=c=\mathrm{const},
]

[
\alpha_1^2-\alpha_2^1=\varepsilon(\alpha_3^4-\alpha_4^3),\qquad
\alpha_1^1+\alpha_2^2+\alpha_3^3+\alpha_4^4=0,
]

[
(\alpha_3^3-\alpha_4^4)^2-(\alpha_4^3-\alpha_3^4)^2=cI,
\tag{2}
]

where (I) is a known invariant of the congruence (the Welsch invariant). Since the frame is completely fixed, (ds) and (\alpha_i^k) are projective invariants of the ruled surface belonging to the given congruence.

§ 2. Computational formulas. Consider the frame of a congruence, referred to the torses and Laplace transformations, introduced in ((^7)), Ch. VI, § 4. Its derivational formulas may be written, in the notation of §§ 185—191 ((^4)) (replacing only (A_i) and (\omega_i^k) by (B_i) and (v_i^k)), in the form:
(dB_i=v_i^kB_k), where (v_1^3) and (v_2^4) are the principal forms, and the remaining (v_i^k) are expressed through them by means of the Fubini coefficients (\alpha,\alpha',\beta,\beta',\gamma), etc.; moreover (\beta=\beta'=\beta_2=\beta_1'=0) by virtue of the choice of the vertices (B_3) and (B_4), and (v_1^1+v_2^2+v_3^3+v_4^4=0) by virtue of the normalization ((B_1B_2B_3B_4)=1). This frame is connected with our frames by the formulas:

[
B_1=\lambda_1(A_1+A_2),\qquad B_2=\lambda_2(A_1-A_2),
]

[
B_3=\lambda_3(A_3+A_4),\qquad B_4=\lambda_4(A_3-A_4),
]

where

[
4\lambda_1\lambda_2\lambda_3\lambda_4=1,\qquad
4\gamma\gamma'\lambda_1\lambda_2=c\lambda_3\lambda_4,\qquad
\gamma'\lambda_1^3\lambda_4^2\lambda_3-\gamma\lambda_2^3\lambda_3^2\lambda_4
+\varepsilon\bigl(\alpha\lambda_1^2\lambda_2\lambda_4^3-\alpha'\lambda_1\lambda_2^2\lambda_3^3\bigr)=0.
]

From this one obtains the following formulas for computing (\alpha_i^k):

[
c\,ds^2=4\gamma\gamma' v_1^3v_2^4,\qquad
4\alpha_1^1\,ds=\psi_1+2\Omega q,\qquad
4\alpha_2^2\,ds=\psi_1-2\Omega q,
]

[
4\alpha_3^3ds=-\psi_1+2\Omega_1q,\qquad
4\alpha_4^4ds=-\psi_1-2\Omega_1q,
]

[
4\alpha_1^2ds=d\ln\varphi_2v_2^4-d\ln\varphi_1v_1^3+2\bigl(v_1^1-v_2^2-\varepsilon\Omega_2q\bigr),
]

[
4\alpha_2^1ds=d\ln\varphi_2v_2^4-d\ln\varphi_1v_1^3+2\bigl(v_1^1-v_2^2+\varepsilon\Omega_2q\bigr),
]

[
4\alpha_3^4ds=d\ln\varphi_2v_1^3+2\bigl(v_3^3-v_4^4-\Omega_2q\bigr)-d\ln\varphi_1v_2^4,
]

[
4\alpha_4^3ds=d\ln\varphi_2v_1^3+2\bigl(v_3^3-v_4^4+\Omega_2q\bigr)-d\ln\varphi_1v_2^4,
]

[
8\alpha_3^1ds^2=4\psi_2+c\psi_3q\,ds^2,\qquad
8\alpha_3^2ds^2=4\psi_2^+c\psi_3^q\,ds^2,
]

[
8\alpha_4^1ds^2=4\psi_2^-c\psi_3^q\,ds^2,\qquad
8\alpha_4^2ds^2=4\psi_2-c\psi_3q\,ds^2,
]

where
[
\psi_1=\gamma_1'v_1^3+\gamma_2v_2^4,\quad
\varphi_1=\gamma(v_2^4)^2-\varepsilon\alpha(v_1^3)^2,\quad
\varphi_2=\gamma'(v_1^3)^2-\varepsilon\alpha'(v_2^4)^2,
]
[
\psi_2=v_1^3v_3^1+v_1^4v_4^3,\quad
\psi_2^*=v_1^3v_3^1-v_2^4v_4^3,\quad
\gamma\gamma'\psi_3=\alpha\beta_1\gamma'(v_1^3)^3-\varepsilon\alpha\alpha'(\beta_1v_2^4+\beta_2'v_1^3)v_1^3v_2^4+\gamma\alpha'\beta_2'(v_2^4)^3,
]

[
\gamma\gamma'\psi_3^*=\alpha'\beta'_2\gamma\,(v_2^4)^3+\alpha\alpha'(\beta_1v_2^4-\beta'_2v_1^3)v_1^3v_2^4-\alpha\beta_1\gamma'(v_1^3)^3,
]
[
\Omega=2\gamma\gamma'(v_1^3)^2(v_2^4)^2-\varepsilon[\alpha\gamma'(v_1^3)^4+\alpha'\gamma(v_2^4)^4],
]
[
\Omega_1+\Omega_2=2\alpha(v_1^3)^2\varphi_2,\qquad
\Omega_1-\Omega_2=2\alpha'(v_2^4)^2\varphi_1
]

are relatively invariant differential forms and (q=(v_1^3v_2^4\varphi_1\varphi_2)^{-1/2}).

These formulas show that the invariants (\alpha_1^2,\alpha_2^1,\alpha_3^4,\alpha_4^3) are of the second order, while the remaining (\alpha_i^k) are of the first order, and also that in the (K)-frame there are excluded from consideration the surfaces ((\varphi_1\varphi_2){\varepsilon=1}=0), along which the line (F_1G_1) (or (F_2G_2)) coincides with an asymptotic tangent, whereas in the (M)-frame there are excluded from consideration the surfaces ((\varphi_1\varphi_2)=0), for which one of the focal lines is asymptotic.

§ 3. Geometric meaning of the invariants. The geometric characterization of the elements of our frames is clear from § 1. We shall call the vertices (A_1) and (A_2) the projective centers of the ray. Consider the points (X_1^i), (X_2^i), (X_3^i), (X_4^i) ((i=1,2)) of intersection of the ray (A_1A_2) with the planes of the pencil having axis (A_3A_4) and passing, respectively, through:

1) points infinitely near to the foci of the ray on the given ruled surface (S);

2) points infinitely near to the centers of the ray on (S);

3) points infinitely near to the centers of the ray on a ruled surface passing through the ray, for which (\alpha_1^2=0) or (\alpha_2^1=0), and which has two common nearby rays with (S);

4) tangents to the lines described by the vertices (A_3) and (A_4) of the frame.

We shall denote by (Y^i) the points of intersection of the line (A_3A_4) with the planes of the pencil having axis the ray (A_1A_2) and passing through points infinitely near to the points (A_i) ((i=3,4)). Then we obtain the following formulas, showing the geometric meaning of our invariants:

[
\begin{aligned}
c\,ds^2&=4DV_0(F_1F_2X_1^1X_2^1),&
\alpha_1^2ds&=DV_0(A_2A_1F_1X_2^1),\
\alpha_2^1ds&=DV_0(A_1A_2F_1X_2^2),&
\alpha_3^4ds&=DV_0(A_4A_3G_1Y^3),\
\alpha_4^3ds&=DV_0(A_3A_4G_1Y^4),&
\alpha_3^2ds^2&=2DV(A_2A_1F_1X_3^1),\
\alpha_4^1ds^2&=2DV_0(A_1A_2F_1X_3^2),&
\alpha_3^1&=\alpha_3^2DV(A_1A_2F_1X_4^1),\
\alpha_4^2&=\alpha_4^1DV(A_2A_1F_1X_4^2),
\end{aligned}
]

where (DV) denotes the compound ratio, and (DV_0) its principal part.

We shall call (E)-surfaces the ruled surfaces belonging to the given congruence and giving an extremum of the invariant (ds). Denote by (\bar\alpha_i^k, \overline{ds}) the invariants of the (E)-surface conjugate to (S). Then we shall have

[
(\alpha_1^1+\alpha_2^2)\,ds=
\begin{cases}
(\bar\alpha_1^2-\bar\alpha_3^4)\,\overline{ds}, & \text{for }\varepsilon=1,\
(\bar\alpha_1^2-\bar\alpha_4^3)\,\overline{ds}, & \text{for }\varepsilon=-1.
\end{cases}
]

These formulas, together with (2), make it possible to determine the geometric meaning of the invariants (\alpha_i^i).

§ 4. Natural equations. The results obtained in §§ 1 and 3 make it possible to find the natural equations of projective-invariant classes of ruled surfaces belonging to the given congruence. We indicate some of them: (\alpha_1^2=0)—surfaces on which the tangent to the first line of centers ((A_1)) coincides with the edge of the frame (A_1A_3); (\alpha_3^4=0)—surfaces for which (A_1A_3) describes a torse whose cuspidal edge does not coincide with the line of centers; (\alpha_3^1=0)—surfaces for which the plane (A_2A_3A_4) contains the tangent to the line ((A_3)); (\alpha_3^2=0)—surfaces for which the plane (A_1A_3A_4) contains the tangent to the line ((A_3)); (\alpha_1^2+\alpha_3^4=0)—surfaces for which the first line of centers

is asymptotic (we note that in the case of an (M)-frame both lines of centers become asymptotic simultaneously); (\alpha_1^2=\alpha_3^4)—surfaces for which (A_1A_3) is an asymptotic tangent (in the case of a (K)-frame, (A_1A_3) and (A_2A_4) possess this property simultaneously); (\alpha_4^3-\alpha_3^4\pm(\alpha_2^2+\alpha_4^4)=0) in a (K)-frame—surfaces one of whose focal lines is asymptotic; (\alpha_4^3-\alpha_3^4\pm(\alpha_4^4+\alpha_1^1)=0) in an (M)-frame—surfaces for which the line (F_1G_1) (or (F_2G_2)) is an asymptotic tangent; (\alpha_4^2\alpha_4^3-\alpha_4^2=0)—surfaces for which (A_1A_4) describes a torse; (\alpha_3^1\alpha_4^2-\alpha_3^2\alpha_4^1=0)—surfaces for which the line (G_1G_2) describes a torse.

Similarly characterized are the surfaces (\alpha_2^1=0,\ \alpha_4^3=0,\ \alpha_4^2=0,\ \alpha_4^1=0), etc. The differential equations of all these surfaces (in torse parameters) are easily obtained from the formulas of § 2.

§ 5. Egorov transformations. Let (v=\mathrm{const}) be a one-parameter family of ruled surfaces belonging to the given congruence and referred to a (K)- or (M)-frame. Let (M\to P(M)) be a projective transformation whose coefficients depend on the parameter (v). We shall seek such families of ruled surfaces belonging to the congruence for which one can find a transformation (M\to P(M)) under which one of the focal surfaces of the congruence (A_1A_2) is transformed into the focal surface of the congruence (P(A_1)P(A_2)) (such transformations we shall call—by analogy with ((^1)) and ((^2))—projective Egorov transformations of the first kind).

The natural equation of the required class (containing (\infty^{15}) surfaces) can be written in the form*

[
\det |q_i^k|=0.
\tag{3}
]

Here (q_1^1=q_1^6=1,\ q_1^2=q_1^5=\pm 1), and the remaining (q_1^k=0). To find (q_2^k,\ldots,q_{15}^k) we obtain the recurrence formulas
[
q_{i+1}^k=\frac{d}{ds}q_i^k+q_i^\alpha\beta_\alpha^k;
]
(\beta_\alpha^k) are the coefficients of the derivative formulas, obtained from (1), (dR_\alpha/ds=\beta_\alpha^kR_k), where

[
R_1=(P'(A_1),\ P(A_1),\ P(A_2),\ P(A_3)),\quad
R_2=(P'(A_1),\ P(A_1),\ P(A_2),\ P(A_4))
]

and so on, while (P') denotes the projective transformation whose coefficients are the derivatives of the coefficients of the transformation (P).

In an analogous way one finds the surfaces by means of which Egorov transformations of the second kind are effected (preserving the conjugacy of the net of ruled surfaces) and of the third kind (the projective centers remain points harmonically separating the foci). In the first case, in the first row of equation (3), the nonzero entries are (q_1^2=-q_1^5=1); in the second case (q_1^2=q_1^5=1).

Buryat-Mongolian State
Pedagogical Institute
named after D. Banzarov

Received
13 II 1956

References

¹ R. N. Shcherbakov, Uch. zap. Buryat-Mongolian Pedagogical Institute, 5, 61 (1954). ² R. N. Shcherbakov, Matem. sbornik, 37 (79), 3, 527 (1955). ³ S. P. Finikov, The Method of Exterior Forms of Cartan, 1948. ⁴ S. P. Finikov, Theory of Congruences, 1950. ⁵ G. Sannia, Ann. di Matem. (III), 15, 143 (1908). ⁶ G. Bol, Math. Zs., 52, 791 (1950). ⁷ S. P. Finikov, Projective-Differential Geometry, 1937.

[
\text{* All indices run through the values from 1 to 15.}
]