Full Text
A. K. RYBNIKOV
ON SYMMETRIC SPACES OF AFFINE CONNECTION OF THE FIRST CLASS
(Presented by Academician P. S. Aleksandrov on 24 IV 1961)
In an \(N\)-dimensional affine space \(R_N\), consider an \(n\)-dimensional surface. To each point \(M\) of this surface attach an \(n\)-dimensional plane \(S\) (the base plane) passing through this point, and an \((N-n)\)-dimensional plane \(\Sigma\) (the supporting plane), having with \(S\) only the single common point \(M\). In the manifold of base planes \(S\), an affine connection is established by projecting a neighboring base plane onto the original one parallel to the original supporting plane \(\Sigma\). The problem of immersing an \(n\)-dimensional space of affine connection \(A_n\) in \(R_N\) consists in finding in \(R_N\) an \(n\)-dimensional surface, equipped with base and supporting planes, on which the connection established by projection would coincide with the connection in \(A_n\). In this formulation the immersion problem was considered in the works \(({}^{1-5})\).
Let \(R_N\) be an affine space of minimal dimension \(N\) into which the given \(n\)-dimensional \(A_n\) can be immersed. The difference \(N-n\) is called the class of \(A_n\). In the present paper we consider symmetric \(A_n\) of the 1st class with nondegenerate Ricci tensor. In this case the base planes are hyperplanes, and the supporting planes are lines.
- In the subsequent arguments we shall have to use the following lemma, which we state here without proof.
Lemma. An \(n\)-dimensional space of affine connection \(A_n\) without torsion, of the 1st class, with nondegenerate Ricci tensor \(R_{ij}\) \((\operatorname{Det}\|R_{ij}\| \ne 0)\), can be realized in \(R_{n+1}\) only on a hypersurface of rank \(n\), under the condition that the tangent hyperplane is chosen as the base hyperplane. In all other realizations the tensor \(R_{ij}\) is degenerate; and if a non-tangent hyperplane is chosen as the base hyperplane, then, in addition, the matrix composed of the components of the curvature tensor
\[ \left\| R^i{}_{jkl} \right\| \tag{1} \]
where \(i\) denotes rows and \(j,k,l\) columns, must necessarily have rank \(1\).
In view of the lemma, we must refer the hypersurface realizing our connection to a frame of the 1st order. By its equipment we shall henceforth understand only the equipment by supporting lines.
- Consider a symmetric \(A_n\) \((n>2)\) of the 1st class, of which, in addition to nondegeneracy of the Ricci tensor \(R_{ij}\), we require that the matrix (1) have rank \(>2\). The realization equations for \(A_n\) in \(R_{n+1}\) have the form
\[ \omega^{n+1}=0;\qquad \omega_i^{n+1}=\Lambda_{ij}\omega^j;\qquad \omega_{n+1}^{\,i}=\Lambda_j^{\,i}\omega^j \]
\[ (\Lambda_{ij}=\Lambda_{ji};\qquad \operatorname{Det}\|\Lambda_{ij}\|\ne 0;\qquad i,j=1,\ldots,n). \tag{2} \]
The components of the curvature tensor of our \(A_n\) are expressed through \(\Lambda_{ij}\) and \(\Lambda_j^{\,i}\) in the following way:
\[ R^i{}_{jkl}=\frac{1}{2}\left(\Lambda_{jk}\Lambda_l^{\,i}-\Lambda_{jl}\Lambda_k^{\,i}\right). \tag{3} \]
Since \(\operatorname{Det}\|\Lambda_{ij}\|\ne 0\), we may introduce into consideration the object \(\Lambda^{ij}\), whose components are the elements of the matrix \(\|\Lambda^{ij}\|\) inverse to the matrix \(\|\Lambda_{ij}\|\). From expressions (3) there follow the relations:
\[ \Lambda^{ls}\Lambda^i_{jkl}=0\quad (s\ne k,l); \tag{4} \]
\[ \Lambda^{jk}\Lambda^i_{ikl}=\frac12\Lambda^i_l\quad \text{(no summation over \(k\)).} \tag{5} \]
Covariant differentiation of relations (4) with respect to the connection of the space \(A_n\) gives, by virtue of (5):
\[ \Lambda^i_l\Lambda^{as}\Lambda_{kam}-\Lambda^i_k\Lambda^{as}\Lambda_{lam}=0\quad (s\ne k,l), \tag{6} \]
where \(\Lambda_{ijk}\) are the components of the fundamental object of the 3rd order of the hypersurface.
From the fact that the matrix (1) has rank \(>2\), it follows that the matrix \(\|\Lambda^i_j\|\) also has rank \(>2\). In view of this, from relations (6) we obtain
\[ \Lambda^{as}\Lambda_{kam}=0\quad (s\ne k). \tag{7} \]
Since \(\Lambda_{ijk}\) are symmetric in all their indices, it follows from equations (7) that
\[ \Lambda_{ijk}=0. \tag{8} \]
The equalities (8) can hold if and only if our hypersurface is a hyperquadric and is furnished with Blaschke affine normals.
Thus, we have proved the following theorem.
Theorem 1. Symmetric \(A_n\) \((n>2)\) of the first class with nondegenerate Ricci tensor, for which the matrix (1) has rank \(>2\), can be realized in \(R_{n+1}\) only on hyperquadrics of rank \(n\), furnished with Blaschke affine normals.
- From the equations obtained by G. F. Laptev (6) for a hyperquadric in projective space, it follows that the equations of a hyperquadric of rank \(n\), furnished with Blaschke affine normals, have the following form:
\[ \omega^{n+1}=0;\qquad \omega^{n+1}_i=\Lambda_{ij}\omega^j;\qquad \omega^i_{n+1}=-\frac{\hat l}{n+2}\omega^i, \tag{9} \]
\[ d\Lambda_{ij}=\Lambda_{ik}\omega^k_j+\Lambda_{kj}\omega^k_i-\Lambda_{ij}\omega^{n+1}_{n+1};\qquad d\hat l=\hat l\,\omega^{n+1}_{n+1}\quad (\operatorname{Det}\|\Lambda_{ij}\|\ne 0). \]
The components of the curvature tensor of the affine connection induced on the hyperquadric by such an equipment, and the components of the Ricci tensor of this connection, have the form:
\[ R^i{}_{jkl}=\frac{\hat l}{2(n+2)}\left(\delta^i_k\Lambda_{jl}-\delta^i_l\Lambda_{jk}\right); \tag{10} \]
\[ R_{ij}=-\frac{\hat l(n-1)}{2(n+2)}\Lambda_{ij}. \tag{11} \]
It is easy to see that this connection is symmetric and equiaffine.
The curvature tensor satisfies the relations
\[ R^i{}_{jkl}=-\frac{1}{n-1}\left(\delta^i_k R_{jl}-\delta^i_l R_{jk}\right). \tag{12} \]
If we introduce into consideration the metric tensor
\[ g_{ij}=-\frac{n}{R}R_{ij},\qquad \text{where } R=\operatorname{const}, \tag{13} \]
then relations (12) take the form
\[ R^i{}_{jkl}=K\left(\delta^i_k g_{jl}-\delta^i_l g_{jk}\right),\qquad \text{where } K=\frac{1}{n(n-1)}R. \tag{14} \]
Consequently, our connection may be regarded as the connection of a Riemannian space of constant curvature.
This allows us to state the following theorem:
Theorem 2. The affine connection induced in \(R_{n+1}\) on a hyperquadric of rank \(n\) by Blaschke affine normals is the connection of a Riemannian space of constant curvature.
- P. A. Shirokov proved that Riemannian spaces of constant curvature are spaces of the 1st class and that there exist no symmetric Riemannian spaces of the 1st class with nondegenerate Ricci tensor, distinct from spaces of constant magnitude. From Theorems 1 and 2 it follows that there exist no non-Riemannian symmetric spaces of the 1st class satisfying the restrictions indicated in Theorem 1. We obtain the following theorem:
Theorem 3. There exist no symmetric spaces of affine connection \(A_n\) \((n>2)\) of the 1st class with nondegenerate Ricci tensor, for which the matrix (1) has rank \(>2\), distinct from Riemannian spaces of constant curvature.
Moscow State University
named after M. V. Lomonosov
Received
21 IV 1961
CITED LITERATURE
\(^{1}\) G. F. Laptev, DAN, 41, No. 8 (1943).
\(^{2}\) G. F. Laptev, DAN, 47, No. 8 (1945).
\(^{3}\) O. Galvani, J. de math. pures et appl., 25, 209 (1946).
\(^{4}\) A. K. Rybnikov, Vestn. Mosk. Univ., ser. matem., mekh., astronom., fiz., khim., No. 5 (1959).
\(^{5}\) A. K. Rybnikov, Vestn. Mosk. Univ., ser. 1, matem. i mekh., No. 1 (1960).
\(^{6}\) G. F. Laptev, Tr. Mosk. matem. obshch., 2, 275 (1953).
\(^{7}\) P. A. Shirokov, Uch. zap. Kazansk. gos. univ., 114, book 8 (1954).