G. I. KRUCHKOVICH

ON RIEMANNIAN SPACES WITH A SUFFICIENTLY LARGE GROUP OF MOTIONS

(Presented by Academician I. G. Petrovskii on 13 IV 1960)

1°. As is known, spaces of constant curvature, and only they, possess the largest possible group of motions \(G_r\) for Riemannian spaces \(V_n\), whose dimension is \(r = n(n + 1)/2\). G. Fubini proved \((^1)\) that no space \(V_n\) can admit a full group of motions of dimension \(r = n(n + 1)/2 - 1\). Later it became clear that in the distribution of the dimensions of full groups of motions of spaces \(V_n\) there are entire gaps. From a number of works \((^{2-7})\) there follows the existence of at least two such gaps. Muto \((^{8,9})\) investigated groups of motions in non-conformally Euclidean spaces and found for them a third gap. However, the question of the existence of a third gap for all spaces \(V_n\) remained open, since conformally Euclidean spaces had not been investigated. The purpose of the present note is to fill this gap and thereby to give the final picture of the distribution of the dimensions of full groups of motions of Riemannian spaces within the limits of the three gaps indicated. At the same time, it gives an enumeration of all Riemannian spaces admitting a transitive or nontransitive group of motions of sufficiently large dimension \((r > (n - 2)(n - 3)/2 + 8)\). As it turns out, all such spaces \(V_n\) for \(n \ne 6\) and \(n \ne 8\) are semi-reducible \((^{10})\). We note that the metric \(ds^2\) of the space \(V_n\) is assumed positive-definite.

2°. We first formulate the theorems following from the works cited above, to which reference will be made below.

Theorem A \((^{2,3})\). A Riemannian space \(V_n\), for \(n \ne 4\), cannot admit a full group of motions \(G_r\) if

\[ \frac{n(n - 1)}{2} + 1 < r < \frac{n(n + 1)}{2} \]

(the 1st gap).

Theorem B \((^{4,7})\). A Riemannian space \(V_n\) cannot admit a full group of motions \(G_r\) of dimension

\[ \frac{(n - 1)(n - 2)}{2} + 5 < r < \frac{n(n - 1)}{2} \]

(the 2nd gap).

Theorem C \((^{4,5,7})\). Every Riemannian space \(V_n\) admitting a group of motions of dimension \(r > (n - 1)(n - 2)/2 + 5\) is a space of constant curvature or a subprojective space of Kagan.

Theorem D \((^{8,9})\). If a non-conformally Euclidean space \(V_n\) \((n > 6, \ne 8)\) admits a group of motions of dimension \(r > (n - 2)(n - 3)/2 + 8\), then necessarily \(r = (n - 1)(n - 2)/2 + k\) \((k = 0, 1, 2, 3)\), and the space is semi-reducible with a metric of the form

\[ ds^2 = ds_0^2(x^1, x^2) + \sigma(x^1, x^2)\, ds_1^2(x^3, \ldots, x^n), \]

where \(ds_1^2\) is an \((n - 2)\)-dimensional metric of constant curvature.

In addition, we shall need the following theorem:

Theorem E \((^{11})\). The full orthogonal group \(O(n)\), for \(n \ne 4\), cannot contain subgroups whose dimension is greater than the dimension of \(O(n - 1)\).

3°. Consider a Riemannian space \(V_n\) admitting a group of motions \(G_r\). Denote by \(H_m\) the stationary subgroup of some point of the space, and by \(\widetilde H_m\) the isotropy group, i.e. the group of rotations in the tangent space \(E_n\) of this point induced by the group \(H_m\). In a joint paper of the author and Gu Chao-hao the following theorem was proved:

Theorem F (\(^{12}\)). Let the group of motions \(G_r\) be transitive, and let the isotropy group \(\widetilde H_m\) decompose into the direct product of two subgroups \(H^{(0)}\) and \(H^{(1)}\), acting on mutually orthogonal complementary planes \(E_{n-q}\) and \(E_q\). If \(H^{(1)}\) is irreducible in \(E_q\) and admits no one-dimensional groups of rotations in \(E_q\) permutable with it, then the space \(V_n\) is semireducible. Its metric can be brought to the form

\[ ds^2=ds_0^2(x^1,\ldots,x^{n-q})+\sigma(x^1,\ldots,x^{n-q})ds_1^2(x^{n-q+1},\ldots,x^n), \tag{1} \]

and the group \(G_r\) is a non-mixing group of motions with respect to the semireducible decomposition (1).

The following supplements may be made to this theorem:

1) If \(\sigma\ne\mathrm{const}\) in (1), then one can show that the metric \(ds_1^2\) is Euclidean, and the decomposition (1) reduces to the form

\[ ds^2=ds_0^2(x^1,\ldots,x^{n-q})+e^{2ax^1}(dx^{n-q+1\,2}+\ldots+dx^{n\,2}). \tag{2} \]

2) The requirement that \(H^{(1)}\) admit no one-dimensional groups \(G_1\) of rotations in \(E_q\) permutable with it is fulfilled automatically if \(q\) is odd. If \(q\) is even, then this requirement is always fulfilled when the dimension of the group \(H^{(1)}\) is greater than \((q/2)^2\). Indeed, as É. Cartan showed, the infinitesimal matrix of the group of rotations \(G_1\), permutable with an irreducible group of orthogonal matrices, is brought to the form \(I=\left\|\begin{matrix}0&E\\-E&0\end{matrix}\right\|\), where \(E\) is the identity matrix of order \(q/2\). Every skew-symmetric matrix permutable with \(I\), as is easily verified, has the form \(L=\left\|\begin{matrix}A&B\\-B&A\end{matrix}\right\|\), where \(A\) is a skew-symmetric and \(B\) a symmetric matrix of order \(q/2\). The maximal number of parameters which the matrix \(L\) can contain is
\[ \tfrac12[q/2(q/2-1)]+\tfrac12[q/2(q/2+1)]=(q/2)^2. \]

4°. Lemma 1. If the isotropy group \(\widetilde H_m\) leaves invariant a \(q\)-dimensional plane \(E_q\), then

\[ r<\frac{n(n+1)}{2}-q(n-q). \]

Indeed, the orthogonal complement \(E_{n-q}\) to \(E_q\) is also invariant with respect to \(\widetilde H_m\), and therefore its dimension \(m\) cannot exceed the sum of the dimensions of the full rotation groups in \(E_q\) and \(E_{n-q}\). Hence the required estimate follows at once, since \(r\le m+n\).

From Lemma 1 the following assertions follow in an obvious way:

Lemma 2. If \(r>(n-2)(n-3)/2+6\), then \(\widetilde H_m\) cannot fix any plane \(E_q\) for \(3\le q\le n-3\).

Lemma 3. Let \(\rho_i\) be the roots of the Ricci tensor \(R_{ij}\) of the space \(V_n\), admitting a group of motions \(G_r\) of dimension \(r>(n-2)(n-3)/2+6\). Then only three cases are possible: 1) \(\rho_1=\rho_2=\ldots=\rho_n\); 2) \(\rho_1\ne\rho_2=\ldots=\rho_n\); 3) \(\rho_1,\rho_2\ne\rho_3=\ldots=\rho_n\).

5°. We shall investigate a Riemannian space \(V_n\) with a transitive group of motions \(G_r\) of dimension \(r>(n-2)(n-3)/2+6\), assuming that \(\widetilde H_m\) is reducible. According to Lemma 2 there are only the following possibilities:

I. \(E_n=E_1\times E_{n-1}\) and \(\widetilde H_m\) is an irreducible group of rotations in \(E_{n-1}\).

II. \(E_n=E_2\times E_{n-2}\), and either \(\widetilde H_m\) itself or its subgroup \(\widetilde H_{m-1}\) is an irreducible group of rotations in \(E_{n-2}\). In this case the dimension of the stationary group satisfies the inequality \(m>(n-3)(n-4)/2+3\).

In case I all the conditions of Theorem F are fulfilled. The absence of interchange rotations from \(\widetilde H_m\) on \(E_{n-1}\) follows from the fact that for \(n>3\) the dimension of \(\widetilde H_m\) is greater than \(((n-1)/2)^2\). Therefore

\[ ds^2=dx^{1\,2}+\sigma(x^1)\,ds_1^2(x^2,\ldots,x^n). \tag{3} \]

The group \(G_r\) is a non-interchanging transitive group of motions in (3). Therefore, for \(\sigma=\mathrm{const}\), \(G_r\) decomposes into the direct product of the group of translations along the lines \(x^1\) and the subgroup \(G_{r-1}\) of motions in \(ds_1^2\). Since
\[ r-1>(n-2)(n-3)/2+5, \]
it follows, by Theorem C, that the space \(ds_1^2\) must have constant curvature \(K\), since in a subprojective space the isotropy group is reducible. Consequently, every space (3) is a subprojectively reducible space for \(K\ne0\), or Euclidean for \(K=0\). If \(\sigma\ne\mathrm{const}\), then one obtains the metric
\[ ds^2=ds^{1\,2}+e^{2ax^1}(dx^{2\,2}+\cdots+dx^{n\,2}), \]
which has constant negative curvature.

Case II is possible only for \(n>6\), since for \(n\leqslant 6\) the dimension of \(\widetilde H_m\) is greater than the dimension of \(O(n-2)\). If \(n>6\), then, by dimension, \(\widetilde H_m\) occupies an intermediate position between \(O(n-2)\) and \(O(n-3)\), and therefore, by Theorem E, the group \(\widetilde H_m\) or its subgroup \(\widetilde H_{m-1}\) coincides with \(O(n-2)\). Consequently,
\[ m=(n-2)(n-3)/2 \]
or
\[ m=(n-2)(n-3)/2+1, \]
and
\[ r=(n-1)(n-2)/2+2 \]
or
\[ r=(n-1)(n-2)/2+3. \]
In both cases the conditions of Theorem F are fulfilled and

\[ ds^2=ds_0^2(x^1,x^2)+\sigma(x^1,x^2)\,ds_1^2(x^3,\ldots,x^n), \tag{4} \]

where \(ds_1^2\) must necessarily have constant curvature \(K_1\). Since the group \(G_r\) is non-interchanging, \(ds_0^2\) admits a group of motions \(G_2\) or \(G_3\), and therefore also has constant curvature \(K_0\). For \(\sigma=\mathrm{const}\), the space \(V_n\) thus decomposes into the direct product of two-dimensional and \((n-2)\)-dimensional spaces of constant curvature, and then
\[ r=(n-1)(n-2)/2+3. \]
For \(\sigma\ne\mathrm{const}\) it is easy to find that the metric reduces to the form

\[ ds^2=dx^{1\,2}+e^{2bx^1}dx^{2\,2}+e^{2ax^1}(dx^{3\,2}+\cdots+dx^{n\,2}),\qquad a\ne0,\quad b\ne0,\quad a\ne b. \tag{5} \]

The group of motions in this case has dimension
\[ r=(n-1)(n-2)/2+2. \]

\(6^\circ\). If the space \(V_n\) is not Einstein, then among the roots of the Ricci tensor there are at least two distinct ones; and since the corresponding eigenspaces in \(E_n\) are invariant with respect to the group of motions, \(\widetilde H_m\) is reducible. In this case, from the arguments of item \(5^\circ\) there follows the theorem:

Theorem 1. If a non-Einstein \(n\)-space \(V_n\) admits a transitive group of motions \(G_r\) of dimension
\[ r>(n-2)(n-3)/2+6, \]
then this space belongs to one of the following types:

1) \(V_n\) of constant curvature, \(r=n(n+1)/2\).

2) \(V_n=V_1\times V_{n-1}\), where \(V_{n-1}\) is of constant curvature \(K\ne0\), \(r=n(n-1)/2+1\).

3) \(V_n=V_2\times V_{n-2}\), where \(V_2\) and \(V_{n-2}\) are of constant curvature,
\[ r=(n-1)\times (n-2)/2+3. \]

4) \(V_n\) is defined by the metric (5),
\[ r=(n-1)(n-2)/2+2. \]

An Einstein space of nonconstant curvature is not conformally Euclidean. Assuming
\[ r>(n-2)(n-3)/2+8\quad (n>6,\ \ne 8) \]
and applying Theorem D to it, we arrive at case 3) of Theorem 1. For \(n=6\) and \(n=8\) the question is still not completely clarified. It is known that in these cases, to the types of spaces listed in Theorem 1, there are added the symmetric \(V_6\) and \(V_8\) belonging to the class of spaces \(V_{2p}\) with group of motions \(G_{p^2+2p}\), having an irreducible isotropy group \(\widetilde H_{p^2}\) \((^{18})\). As a result one obtains:

Theorem 2. Every Riemannian space \(V_n\) admitting a transitive group of motions of dimension
\(r>(n-2)(n-3)/2+8\) \((n>6,\ne 8)\) belongs to one of the types indicated in Theorem 1.

We note that, using Lemma 3 and the condition of constancy of the Ricci roots \(\rho_i\), which follows from the transitivity of \(G_r\), one can prove Theorem 1 specifically for conformally Euclidean spaces (without the Einstein condition). In this case case 4) drops out, since it gives a non-conformally Euclidean space.

\(7^\circ\). Let us now turn to spaces \(V_n\) with a nontransitive group of motions \(G_r\) of dimension
\(r>(n-2)(n-3)/2+6\) \((n>4)\). In this case the orbits of the group may be:

I. Geodesically parallel hypersurfaces \(V_{n-1}\) \((n>4)\).
II. Surfaces \(V_{n-2}\) \((n>8)\).

In case I, if the isotropy group \(\widetilde H_m\) is irreducible in \(E_{n-1}\), the tangent space to \(V_{n-1}\), then it can be shown that the space \(V_n\) is subprojective. Its metric has the form (3), where \(ds_1^2\) is a metric of constant curvature, and \(\sigma\ne \mathrm{const}\). If the group \(\widetilde H_m\) is reducible in \(E_{n-1}\), then as a result we obtain

\[ ds^2=dx^{1^2}+\varphi(x^1)\,dx^{2^2}+\sigma(x^1)\,ds_1^2(x^3,\ldots,x^n), \tag{6} \]

where \(ds_1^2\) is a metric of constant curvature, and the dimension of the group \(G_r\) is
\((n-1)(n-2)/2+1\).

In case II the group \(\widetilde H_m\) is irreducible in \(E_{n-2}\), the tangent space to the orbit \(V_{n-2}\), and as a result we obtain the metric (4), where \(ds_1^2\) has constant curvature, and
\(r=(n-1)(n-2)/2\).

Thus we obtain the theorem:

Theorem 3. If the space \(V_n\) admits a nontransitive group of motions \(G_r\) of dimension
\(r>(n-2)(n-3)/2+6\) \((n>4)\), then it belongs to one of the following types:

1) \(V_n\) is subprojective, \(r=n(n-1)/2\), the orbits of the group are \(V_{n-1}\).

2) \(V_n\) is defined by the metric (6), where \(ds_1^2\) has constant curvature,
\(r=(n-1)(n-2)/2+1\), the orbits of the group are \(V_{n-1}\).

3) \(V_n\) is defined by the metric (4), where \(ds_1^2\) has constant curvature,
\(r=(n-1)(n-2)/2\), the orbits of the group are \(V_{n-2}\).

\(8^\circ\). From the preceding results the following conclusions also follow:

Theorem 4. Every Riemannian space admitting a transitive or nontransitive group of motions \(G_r\) of dimension
\(r>(n-2)(n-3)/2+8\) \((n\ne 6,8)\) is a semireducible space.

Theorem 5. A Riemannian space \(V_n\) cannot admit a complete group of motions \(G_r\) whose dimension satisfies one of the inequalities:

\[ \text{1) }\quad \frac{n(n-1)}{2}+1<r<\frac{n(n+1)}{2}\quad (n\ne 4); \]

\[ \text{2) }\quad \frac{(n-1)(n-2)}{2}+3<r<\frac{n(n-1)}{2}\quad (n\ne 6,8); \]

\[ \text{3) }\quad \frac{(n-2)(n-3)}{2}+8<r<\frac{(n-1)(n-2)}{2}. \]

All-Union Correspondence
Power Engineering Institute

Received
7 IV 1960

CITED LITERATURE

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