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).