Reports of the Academy of Sciences of the USSR

1. Volume 133, No. 1

MATHEMATICS

V. V. GOLDBERG

A FAMILY OF AXIAL PLANES OF A TWO-DIMENSIONAL SURFACE OF FOUR-DIMENSIONAL PROJECTIVE SPACE AND SOME PROBLEMS CONNECTED WITH THIS FAMILY

(Presented by Academician P. S. Aleksandrov on 2 III 1960)

1. To each point \(A_0\) of a two-dimensional surface \((A_0)\) of four-dimensional projective space we attach a projective frame formed by five analytic points: the point \(A_0\), the points \(A_1,A_4\) and \(A_2,A_3\), of which the first two are the points corresponding to the point \(A_0\) under two successive Laplace transformations of the surface \((A_0)\) in one direction, and the other two—in the other direction. The infinitesimal displacements of the vertices of the frame are defined by the equations \(dA_i=\omega_i^j A_j\), where the forms \(\omega_i^j\) satisfy the structure equations of projective space (1): \(D\omega_i^j=[\omega_i^k\omega_k^j]\) \((i,j,k=0,1,2,3,4)\).

With the indicated choice of the vertices of the frame we shall have

\[ \begin{gathered} \omega^3=0,\quad \omega^4=0,\quad \omega_1^2=0,\quad \omega_2^1=0,\quad \omega_1^3=0,\quad \omega_2^4=0,\\ \omega_1^4=a_1^4\omega^1,\quad \omega_2^0=a_2^0\omega^2,\quad \omega_4^3=a_4^3\omega^1,\quad \omega_4^0=a_4^0\omega^1,\quad \omega_4^2=a_4^2\omega^1,\\ \omega_2^3=b_2^3\omega^2,\quad \omega_1^0=b_1^0\omega^2,\quad \omega_3^4=b_3^4\omega^2,\quad \omega_3^0=b_3^0\omega^2,\quad \omega_3^1=b_3^1\omega^2,\\ \omega_4^1=a_4^1\omega^1+b_4^1\omega^2,\quad \omega_3^2=a_3^2\omega^1+b_3^2\omega^2. \end{gathered} \tag{1} \]

Continuation of these equations gives

\[ \begin{gathered} da_1^4+a_1^4(\omega_0^0-2\omega_1^1+\omega_4^4)=a_{11}^4\omega^1,\\ da_2^0+a_2^0(2\omega_0^0-\omega_1^1-\omega_2^2)=a_{21}^0\omega^1,\\ da_4^1+a_4^1(\omega_0^0-\omega_4^4)=a_{41}^1\omega^1+(a_{42}^1-a_4^3b_3^1)\omega^2,\\ da_4^0+a_4^0(2\omega_0^0-\omega_1^1-\omega_4^4)+(a_4^1b_1^0+a_4^3b_3^0)\omega^2=a_4^1\omega^1,\\ da_4^2+a_4^2(\omega_0^0-\omega_1^1+\omega_2^2-\omega_4^4)+(a_4^0+a_4^3b_3^2)\omega^2=a_{41}^2\omega^1,\\ db_4^1+b_4^1(\omega_0^0+\omega_1^1-\omega_2^2-\omega_4^4)=a_{42}^1\omega^1+b_{42}^1\omega^2 \end{gathered} \tag{2} \]

and also 6 equations obtained from equations (2) by the replacement \(\begin{pmatrix}1&3&a\\ 2&4&b\end{pmatrix}\).

1. We shall call an axial plane of the surface \((A_0)\) at the given point \(A_0\) a two-dimensional plane along which the three-dimensional osculating planes of the lines of the unique conjugate net of the surface \((A_0)\) passing through the point \(A_0\) intersect. Thus, with every surface of four-dimensional projective space there is associated a two-parameter family of axial planes. The axial planes of the surfaces \((A_0)\) and \((A_1)\) at the points \(A_0\) and \(A_1\) are, respectively, the planes

\[ \pi_0=[A_0,\; a_4^2A_2+a_4^3A_3,\; b_3^1A_1+b_4^3A_4], \]

\[ \pi_1=[A_1,\; a_4^0A_0+a_4^2A_2+a_4^3A_3,\; \alpha_2A_0+\alpha_1A_2], \tag{3} \]

where

\[ \alpha_1=a_4^3a_3^2+a_{41}^2-\frac{a_{41}^3a_4^2}{a_4^3},\qquad \alpha_2=a_4^2a_2^0+r_{41}^0-\frac{a_{41}^3a_4^0}{a_4^3}. \tag{4} \]

The axial plane \(\pi_2\) of the surface \((A_2)\) at the point \(A_2\) is obtained from \(\pi_1\) by the indicated substitution. The points

\[ a_4^2 A_2 + a_4^3 A_3, \qquad b_3^1 A_1 + b_3^4 A_4 \tag{5} \]

are the points of intersection of the axial plane \(\pi_0\), respectively, with the tangent to the line \(\omega^1=0\) on the surface \((A_2)\) and to the line \(\omega^2=0\) on the surface \((A_1)\).

The points of intersection of the planes \(\pi_0\) and \(\pi_1\), \(\pi_0\) and \(\pi_2\), are, respectively, the points

\[ M_{01}=a_4^0 A_0+a_4^2 A_2+a_4^3 A_3, \qquad M_{0,-1}=b_3^0 A_0+b_3^1 A_1+b_3^4 A_4 . \tag{6} \]

These points describe surfaces, which we shall call surfaces associated with the surface \((A_0)\). We shall also say that the surface \((M_{01})\) is associated with the congruence \((A_0A_1)\), and the point \(M_{01}\) is associated with the line \([A_0A_1]\). We shall denote the Laplace transformations of the surface \((A_0)\) in the direction \(\omega^2=0\) by \((A_1),(A_2),\ldots\), and in the direction \(\omega^1=0\) by \((A_{-1}),(A_{-2}),\ldots\).

Theorem 1. The point associated with the line \([A_i,A_{i+1}]\) is the intersection of the tangent planes of the surfaces \((A_{i-1})\) and \((A_{i+2})\) at the points \(A_{i-1}\) and \(A_{i+2}\).

1. The foci of the axial plane \((\pi_0)\) form a curve of the second order. The equation of this focal curve with respect to the coordinate triangle determined by the point \(A_0\) and the points (5) is written in the form

\[ x_0^2-a_4^0x_0x_1-b_3^0x_0x_2+(a_4^0b_3^0-\alpha_1\beta_2)x_1x_2=0, \tag{7} \]

where \(\beta_2\) is obtained from \(\alpha_1\) by the usual substitution.

Theorem 2. The points (5) and (6) lie on the focal curve (7). The latter means that the axial planes of two neighboring Laplace transformations intersect at a point lying on the focal curves of the axial planes of these Laplace transformations.

Theorem 3. The point \(M_0\) is obtained as the intersection of the axial plane \(\pi_0\) with an infinitely near axial plane under displacement \(\omega^1=0\) along the surface \((A_0)\), and as the intersection of the axial plane \(\pi_1\) with an infinitely near one under displacement \(\omega^2=0\) along the surface \((A_1)\).

1. Surfaces with a focal curve of the axial plane decomposing into a pair of distinct straight lines form two classes, each of which depends on one function of two arguments. For surfaces of the first class \((\alpha_1=0)\) the focal curve decomposes into the straight lines \([a_4^2A_2+a_4^3A_3,\ M_{0,-1}]\), \([b_3^1A_1+b_3^4A_4,\ M_{01}]\). The first straight line is obtained as the intersection of the axial plane \(\pi_0\) with an infinitely near axial plane under displacement \(\omega^2=0\) along the surface \((A_0)\), and under all other displacements along this surface points of the second straight line are obtained. The line \(\omega^2=0\) on the surface \((A_0)\) is space, on \((A_1)\) the corresponding line is planar, on \((A_4)\) it is a straight line, and the surface \((A_4)\) itself is developable. The Laplace transformation following \((A_4)\) degenerates into a line—the edge of regression of the surface \((A_4)\). The tangent to the line \(\omega^2=0\) on the surface \((a_4^2A_2+a_4^3A_3)\) passes through the point \(A_0\). The axial plane of the surface \((A_1)\) degenerates into the straight line \([A_1M_{01}]\), carrying two foci: \(M_{01}\) and \(a_3^2 b_3^0 A_1-b_1^0 M_{01}\).

The subclass of the first class \((\alpha_1=\beta_2=0)\), depending on 10 functions of one argument, is formed by surfaces for which the indicated geometric characteristic holds both in the direction \(\omega^2=0\) and in the direction \(\omega^1=0\). The focal curve of the axial plane \(\pi_0\) in this case decomposes into the same pair of straight lines, but now the first is obtained as the intersection of the axial plane \(\pi_0\) with an infinitely near one under displacement \(\omega^2=0\), and the second—under displacement \(\omega^1=0\) along the surface \((A_0)\); under all

under other displacements along this surface, the axial plane \(\pi_0\) intersects the infinitely close axial planes at the point
\(T=a_4^0b_3^0A_0+a_4^0(b_3^1A_1+b_3^4A_4)+b_3^0(a_4^2A_2+a_4^3A_3)\). This point \(T\) describes a surface \((T)\), enveloped by the axial planes of the surface \((A_0)\).

For surfaces of the second class, characterized by the condition \(a_4^3b_3^0-\alpha_1\beta_2=0\), the focal curve decomposes into the straight lines \([a_4^2A_2+a_4^3A_3,\ b_3^1A_1+b_3^4A_4]\), \([M_{01}, M_{0,-1}]\). The first straight line is obtained as the intersection of the plane \(\pi_0\) with the infinitely close axial plane under the displacement \(\alpha_1\omega^1-a_4^0\omega^2=0\) along the surface \((A_0)\), the second—under displacements \(x_1\alpha_1\omega^1+x_2b_3^0\omega^2=0\). The first straight line carries one focus, and the second—two.

Surfaces for which the focal curve of the axial plane splits into a pair of coincident straight lines are characterized by the conditions \(a_4^0=b_3^0=\alpha_1=0\) and exist with an arbitrariness of 8 functions of one argument. In this case the points (5) coincide with the points (6). The straight line determined by them serves as the focal curve of the axial plane \(\pi_0\) and is obtained as the intersection of this plane with the infinitely close one under the displacement \(\omega^2=0\) along the surface \((A_0)\). Under any other displacement one obtains the point \(A_3\), i.e. the infinitely close axial planes of the surface \((A_0)\) form a pencil with center at the point \(A_3\). The point \(A_3\) lies in the axial plane \(\pi_0\), while the surface described by this point degenerates into a line, the tangent to which passes through the point \(b_3^1A_1+b_3^4A_4\), serving as the second focus of this tangent. The line \((A_3)\) serves as the edge of regression of the developable surface described by the point \(b_3^1A_1+b_3^4A_4\). Surfaces of this kind also possess all the properties of surfaces of the first class \((\alpha_1=0)\).

A special case of such surfaces \((a_4^0=b_3^0=\alpha_1=\beta_2=0)\) are surfaces generating a closed five-term Laplace sequence, in which the surfaces \((A_3)\) and \((A_4)\) degenerate into one and the same straight line \(A_3A_4\), serving as the focal curve of the axial plane \(\pi_0\) and remaining fixed under any displacements along this surface \((A_0)\). Such surfaces depend on 6 functions of one argument.

1. Surfaces for which the associated surfaces are related by a Laplace transformation will be called surfaces \(L\). In order that the surfaces \((M_{01})\) and \((M_{0,-1})\), associated with the surface \((A_0)\), be related by a Laplace transformation, it is necessary and sufficient that the equalities

\[ a_2^0b_3^2+a_4^0b_3^4=0,\qquad a_4^1b_1^0+a_4^3l_3^0=0. \tag{8} \]

hold.

In this case the conjugate nets of the surfaces \((M_{01})\) and \((M_{0,-1})\) correspond to the conjugate net of the surface \((A_0)\).

Theorem 4. Surfaces \(L\) exist with an arbitrariness of 12 functions of one argument.

Theorem 5. All focal surfaces of the Laplace sequence generated by a surface \(L\) are surfaces \(L\).

The Laplace sequence generated by a surface \(L\) will be called a sequence \(L\). By virtue of Theorem 5, the surfaces associated with the surfaces of the sequence \(L\) themselves form a Laplace sequence, which we shall call the associated sequence of the original sequence.

Theorem 6. The Laplace sequence associated with a sequence \(L\) is itself a sequence \(L\).

The proof of Theorems 5 and 6 is based on verifying conditions analogous to equalities (8) for the surfaces \((A_1)\) and \((M_{01})\).

1. Let us note three special cases of surfaces \(L\).

A. \(a_4^0=b_3^2=0,\quad a_4^1 b_1^0+a_4^3 b_3^0=0\). The arbitrariness of existence is equal to 11 functions of one argument. For such surfaces, the tangents to the line \(\omega^2=0\) on the surface \((A_2)\) and to the line \(\omega^1=0\) on the surface \((A_{-2})\) intersect, respectively, the planes \([A_1A_{-1}A_{-2}]\) and \([A_0A_1A_2]\); the line \([A_0M_{01}]\) is tangent to the focal curve of the axial plane \(\pi_0\), and both of the indicated properties are preserved for all surfaces of the Laplace sequence.

B. \(a_4^0=b_3^0=a_4^1=b_3^2=0\). The arbitrariness of existence is equal to 10 functions of one argument. The tangent to the line \(\omega^2=0\) on the surface \((A_2)\) intersects the line \(A_1A_{-2}\), while the tangent to the line \(\omega^1=0\) on the surface \((A_{-2})\) intersects the line \(A_1A_2\). The lines \(A_0M_{01}\) and \(A_0M_{0,-1}\) are tangent to the focal curve of the axial plane \(\pi_0\). These properties hold for any surface of the Laplace sequence. The first shows that the original surface is a generalized Slotnik surface \({}^{(1)}\).

Both in case A and in case B, the associated Laplace sequence is of the same type as the original one and is inscribed in it.

C. \(a_4^0=b_3^0=a_4^1=b_3^2=a_4^2=b_3^1=0\). We have the most general closed five-membered Laplace sequence, depending on 8 functions of one argument. Its associated sequence coincides with it itself.

Moscow City Pedagogical Institute
named after V. P. Potemkin

Received
1 III 1960

CITED LITERATURE

\({}^{1}\) S. P. Finikov, The Method of Exterior Forms, Moscow–Leningrad, 1948.