V. D. Tretyakov

On the Question of Line Geometry in Three-Dimensional Klein Spaces

(Presented by Academician A. I. Mal’tsev on 27 IV 1963)

In the present paper a correspondence is established between symmetric conformally Euclidean spaces ($SC$-spaces) of four dimensions and zero signature and the line geometries of various subgroups of projective transformations of three-dimensional space.

1. Consider the collineation

\[ \tilde{x}^{\alpha}=\gamma_{\sigma}^{\alpha}x^{\sigma} \qquad (\alpha,\beta,\ldots,\sigma=1,2,\ldots,n+2); \tag{1} \]

in the projective space \(P_{n+1}\), satisfying the condition

\[ \gamma_{\sigma}^{\alpha}\gamma_{\beta}^{\sigma}=\varepsilon\delta_{\beta}^{\alpha} \qquad (\varepsilon=\pm1,\;0). \tag{2} \]

We shall call this collineation absolute.

The generalized biplanar space \(B_{n+1}\) is the \((n+1)\)-dimensional space whose fundamental group is isomorphic to the subgroup of projective transformations taking the absolute collineation (1) into itself.

For \(\varepsilon=1\) (hyperbolic \(B_{n+1}\)) the absolute collineation is a projective symmetry in an \(m\)-pair; the case \(\varepsilon=-1\) is possible only in a space of an odd number of dimensions (elliptic \(B_{2k+1}\)) and corresponds to a biplanar involution of elliptic type; for \(\varepsilon=0\) (parabolic \(B_{n+1}\)) the matrix \((\gamma_{\beta}^{\alpha})\) is a nilpotent matrix of general form \(({}^{6},\ p.\ 16)\).

A \(B\)-quadric \(Q_n\) of the generalized biplanar space \(B_{n+1}\) is called \(({}^{6},\ p.\ 17)\) a quadric \(Q_n\), \(a_{\alpha\beta}x^{\alpha}x^{\beta}=0\), defined by a symmetric tensor \(a_{\alpha\beta}\), for which the tensor adjoint to it \(({}^{2},\ p.\ 145)\)

\[ b_{\alpha\beta}=a_{\alpha\sigma}\gamma_{\beta}^{\sigma} \tag{3} \]

is also symmetric.

The absolute planes of the collineation (1) are polar conjugate with respect to the \(B\)-quadric in the elliptic and hyperbolic cases and belong to it in the parabolic case.

1. B. A. Rosenfeld has proved \(({}^{3},\ p.\ 368)\) that the groups of motions of the spaces of constant curvature \(S_3\), \({}^{1}S_3\), and \({}^{2}S_3\) are isomorphic to subgroups of motions of the space \({}^{3}S_5\) leaving fixed two planes that are polar conjugate with respect to the absolute, i.e., taking into themselves certain involutions in \(P_5\). Thus, these groups are isomorphic to subgroups of biplanar motions preserving the polarity induced by the \(B\)-quadric. It is not difficult to show that the groups of motions of the Euclidean \(R_3\) and pseudo-Euclidean \({}^{1}R_3\) spaces are isomorphic to subgroups of motions of a biplanar space of parabolic type, preserving a \(B\)-polarity.

In connection with this there arises the question of classifying \(B\)-quadrics and establishing a correspondence between their types and the line geometries of three-dimensional spaces.

1. We shall carry out the classification of \(B\)-quadrics separately for each of the three types \(B_{n+1}\).

A. Hyperbolic type: \(\gamma_\sigma^\alpha \gamma_\beta^\sigma=\delta_\beta^\alpha\). The matrices of the absolute involution \((\gamma_\beta^\alpha)\), of the \(B\)-quadric \((a_{\alpha\beta})\), and of the \(B\)-motion \((t_\beta^\alpha)\) are reduced to the form:

\[ (\gamma_\beta^\alpha)= \begin{pmatrix} -E_{m+1} & 0\\ 0 & E_{n-m+1} \end{pmatrix}; \qquad (a_{\alpha\beta})= \begin{pmatrix} -E_p & & \\ & E_r & 0\\ & 0 & -E_q\\ & & E_s \end{pmatrix}; \qquad (t_\beta^\alpha)= \begin{pmatrix} P_{m+1} & 0\\ 0 & Q_{n-m+1} \end{pmatrix}, \]

where \(E_i\) is the identity matrix of order \(i\); \(P_{m+1}\) and \(Q_{n-m+1}\) are arbitrary (square) matrices. The numbers \(p,q,r,s\) are related by the relations \(p+r=m+1\), \(q+s=n-m+1\). The index \(l\) ([3], p. 297) of the \(B\)-quadric is equal to \(l=p+q\). If \(m=n/2\) (\(n\) even), one can arrange that \(p>q\).

The numbers \(p,q\), and \(m\) form a complete system of invariants of the \(B\)-quadric. If \(n=2k,\ m=k\) (the proper biplanar space), one can introduce a canonical coordinate system in which

\[ (\gamma_\beta^\alpha)= \begin{pmatrix} 0 & E\\ E & 0 \end{pmatrix}. \]

In this case the matrix of a \(B\)-quadric of zero signature can be reduced to the form

\[ (a_{\alpha\beta})= \begin{pmatrix} 0 & C\\ C & 0 \end{pmatrix}; \qquad C= \begin{pmatrix} E_p & 0\\ 0 & E_{k-p+1} \end{pmatrix}. \tag{4} \]

Generally speaking, in this case, in the canonical coordinate system the matrix of a \(B\)-quadric cannot be reduced either to the form (4) or to diagonal form (the latter is possible when \(p=q\)).

B. Elliptic type: \(\gamma_\sigma^\alpha\gamma_\beta^\sigma=-\delta_\beta^\alpha\). The matrices \((\gamma_\beta^\alpha)\), \((a_{\alpha\beta})\), and \((t_\beta^\alpha)\) are reduced to the form:

\[ (\gamma_\beta^\alpha)= \begin{pmatrix} 0 & -E_{k+1}\\ E_{k+1} & 0 \end{pmatrix}; \qquad (a_{\alpha\beta})= \begin{pmatrix} 0 & E_{k+1}\\ E_{k+1} & 0 \end{pmatrix}; \qquad (t_\beta^\alpha)= \begin{pmatrix} P_{k+1} & -Q_{k+1}\\ Q_{k+2} & P_{k+1} \end{pmatrix} \]

(cf. ([1], p. 93)).

C. Parabolic type: \(\gamma_\alpha^\sigma\gamma_\sigma^\beta=0\). The matrices \((\gamma_\beta^\alpha)\), \((a_{\alpha\beta})\), and \((t_\beta^\alpha)\) are reduced to the form

\[ (\gamma_\beta^\alpha)= \begin{pmatrix} 0 & 0 & 0\\ E_r & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}; \qquad (a_{\alpha\beta})= \begin{pmatrix} 0 & C_r & 0\\ C_r & 0 & 0\\ 0 & 0 & B_s \end{pmatrix}; \qquad (t_\beta^\alpha)= \begin{pmatrix} P_r & 0 & 0\\ Q_r & P_r & T\\ S & 0 & R_s \end{pmatrix}; \]

\[ C_r= \begin{pmatrix} -E_k & 0\\ 0 & E_{r-k} \end{pmatrix}; \qquad B_s= \begin{pmatrix} -E_l & 0\\ 0 & E_{s-l} \end{pmatrix}; \]

\(P_r,Q_r,R_s\) are square matrices, and \(S\) and \(T\) are arbitrary rectangular matrices.

A \(B\)-quadric of zero signature exists in \(B_{2n+1}\) in any of the indicated cases; moreover, if the matrix of the absolute collineation is changed in the corresponding manner, the matrix of the \(B\)-quadric will have the form

\[ \begin{pmatrix} 0 & E\\ E & 0 \end{pmatrix}. \]

The results obtained make it possible to carry out a classification of the subgroups of biplanar motions that leave invariant the polarity determined by the given \(B\)-quadric.

1. A. P. Shirokov ([6]) proved that the geometry of any symmetric conformally Euclidean space (an \(SC\)-space) can be realized as the internal geometry of a \(B\)-quadric normalized by means of an absolute involution.

Since the transformation group \(P_5\) leaving invariant the Plücker hyperquadric \(Q_4\) is isomorphic to the group of projective transformations in \(P_3\), the classification of \(B\)-motions leaving \(Q_4\) invariant makes it possible to distinguish all subgroups of projective transformations of three-dimensional space whose line geometries, under the mapping onto the Plücker hyperquadric, determine a symmetric space.

As a result we obtain Table 1. Cases 4–8 of this table correspond to the spaces considered by A. P. Norden ([2], p. 153). Clas-

Table 1

sification of \(SC\)-spaces agrees with the results of P. A. Shirokov \((^7)\). To the line geometry of all the spaces indicated in the last column of Table 1, the results of the work \((^4)\) are applicable. Using the projective interpretation of \(SC\)-spaces given by A. P. Shirokov \((^6)\), it is not difficult to construct a conformal interpretation of all these line geometries.

In this case the absolute invariant of two adjacent straight lines is determined by the quadratic form:

\[ g_{ij}=\partial_i x\,\partial_j x \qquad (i,j=1,\ldots,4), \]

where, as usual, \(xy=a_{\alpha\beta}x^\alpha x^\beta\) \((\alpha,\beta=1,\ldots,6)\), and the normalization of the points (straight lines) \(x\) is subject to the condition \(xx=1\) (see \((^4)\) and \((^6)\), p. 17).

Kuibyshev State
Pedagogical Institute

Received
21 IV 1963

CITED LITERATURE

\(^1\) G. E. Izotov, Izv. vyssh. uchebn. zaved., Matematika, No. 1 (2), 89 (1958).
\(^2\) A. P. Norden, Izv. vyssh. uchebn. zaved., Matematika, No. 4 (17), 145 (1960).
\(^3\) B. A. Rozenfeld, Non-Euclidean Geometries, 1957.
\(^4\) V. D. Tretyakov, Volzhskii Mat. Sborn., No. 1 (1963).
\(^5\) A. P. Shirokov, Uch. zap. Kazansk. gos. univ., 114, book 2 (1954).
\(^6\) A. P. Shirokov, Uch. zap. Kazansk. gos. univ., 116, book 1, 15 (1956).
\(^7\) P. A. Shirokov, Izv. Kazansk. fiz.-matem. obshch., 11, ser. 3, 9 (1938).