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MATHEMATICS
G. I. KRUCHKOVICH and GU CHAO-HAO
A CRITERION FOR SEMI-REDUCIBILITY OF HOMOGENEOUS RIEMANNIAN SPACES
(Presented by Academician P. S. Aleksandrov, 15 II 1958)
Consider a proper Riemannian \((ds^2 > 0)\) homogeneous space \(V_n\) with a continuous group of motions \(G_r\) and the stationary subgroup \(H\) of a given point \(M\). Suppose that \(H\), regarded as a group of rotations in the Euclidean tangent space \(E_n\) at the point \(M\), decomposes into the direct product of two subgroups:
\[
H = H_0 \times H_1,
\]
acting, respectively, on the planes \(E_q\) and \(E_{n-q}\), which are, obviously, mutually orthogonal*.
The purpose of the present note is to establish to what extent, from the indicated decomposition of the stationary group \(H\), there follows a foliation of the whole space \(V_n\) into mutually complementary and orthogonal surfaces \(V_q\) and \(V_{n-q}\).
The following theorem, proved by Wakakuwa, is known \((^1)\):
Theorem 1. If \(H_0\) fixes no one-dimensional direction in the plane \(E_q\), and \(H_1\) fixes none in \(E_{n-q}\), then the space \(V_n\) is reducible:
\[
V_n = V_q \times V_{n-q},
\]
i.e. it is foliated into mutually orthogonal totally geodesic surfaces \(V_q\) and \(V_{n-q}\).
We shall show that, under certain restrictions on the group \(H\), the space \(V_n\) is semi-reducible, i.e. it decomposes into mutually orthogonal surfaces \(V_q\) and \(V_{n-q}\), of which some are totally geodesic, while the others are umbilical and similar to one another \((^2)\). Here it is essential that the restrictions are imposed on only one of the components of the group \(H\); the other remains completely arbitrary.
Choose at the point \(M\) an orthonormal frame
\[
R_0(M,\bar e_i,\bar e_\alpha)^{**}
\]
so that the \(\bar e_i\) form a basis in \(E_q\), and the \(\bar e_\alpha\) in \(E_{n-q}\). Subjecting the frame \(R_0\) to all possible transformations of the group \(G_r\), we obtain a family of frames \(R\), depending on the \(n\) coordinates of the point \(M\) and on the \(r-n\) parameters of the group \(H\), and invariant with respect to \(G_r\). The connection of the space \(V_n\), referred to these frames, as is known, is determined by the formulas
\[
dM = \omega^a \bar e_a,\qquad
d\bar e_a = \omega_a^b \bar e_b
\quad (\omega_a^b+\omega_b^a=0),
\]
where the forms \(\omega^a\) contain only the differentials \(dx^a\). Moreover, in view of the invariance of the planes \(E_q\) and \(E_{n-q}\) with respect to rotations from \(H\), one must have
\[
\omega_i^\alpha = C_{ia}^{\alpha}\omega^a
\quad
(C_{i\alpha}^{\alpha}=-C_{\alpha\alpha}^{i})
\tag{1}
\]
* The case is not excluded in which one of the subgroups, for example \(H_0\), is trivial, i.e. coincides with the identity transformation on the plane \(E_q\).
** The indices everywhere range over the following limits: \(a,b,c=1,\ldots,n;\ i,j,k=1,\ldots,q;\ \alpha,\beta,\gamma=q+1,\ldots,n\).
with constant \(C_{ia}^{\alpha}\) (since \(\omega_i^{\alpha}, \omega^a\) are invariants with respect to the transitive group \(G_r\)).
Under rotations of the group \(H\), the forms \(\omega^a, \omega_b^a\) receive increments:
\[ \delta \omega^a=L_b^a \omega^b\delta t,\qquad \delta \omega_b^a=(L_c^a\omega_b^c-L_b^c\omega_c^a)\delta t, \]
where \(L=\|L_b^a\|\) is one of the matrices of infinitesimal rotations. Applying these formulas to (1) separately for rotations \(L\) from \(H_0\) and \(L\) from \(H_1\), we obtain:
\[ \begin{array}{ll} \text{a) } \underset{1}{L_\beta^\alpha} C_{ik}^{\beta}=0, & \text{b) } \underset{1}{L_\beta^\alpha} C_{i\gamma}^{\beta} = C_{i\beta}^{\alpha}\underset{1}{L_\gamma^\beta}; \\[6pt] \text{c) } -\,\underset{0}{L_i^j} C_{jk}^{\alpha} = C_{ij}^{\alpha}\underset{0}{L_k^j}, & \text{d) } \underset{0}{L_i^j} C_{j\gamma}^{\alpha}=0. \end{array} \tag{2} \]
Let us further assume that one of the components of the group \(H\), for example \(H_1\), is irreducible. In particular, \(H_1\) fixes no one-dimensional directions in \(E_{n-q}\), and therefore from equality a) it follows that \(C_{ik}^{\beta}=-C_{\beta k}^{i}=0\), while c) is identically satisfied. If we denote \(C_k=\|C_{k\beta}^{\alpha}\|\), then b) in (2) can be written briefly as
\[ \underset{1}{L}C_k=C_k\underset{1}{L}. \tag{3} \]
Lemma. If a matrix \(C\) commutes with an irreducible group of orthogonal matrices, then its symmetric part is a scalar matrix (i.e., of the form \(\lambda E\)), and, in the case when the dimension of the space is odd, its skew-symmetric part is the zero matrix.
On the basis of this lemma the elements of the matrix \(C_k\) have the form:
\[ C_{k\beta}^{\alpha}=A_{k\beta}^{\alpha}+\lambda_k\delta_\beta^\alpha,\qquad A_{k\beta}^{\alpha}=-A_{k\alpha}^{\beta},\qquad \delta_\beta^\alpha= \begin{cases} 0, & \alpha\ne\beta,\\ 1, & \alpha=\beta. \end{cases} \]
Using these formulas in the structure equations of the space \(V_n\) under consideration, we obtain:
\[ D\omega^i=[\omega^k\omega_k^i]-A_{i\beta}^{\alpha}[\omega^\alpha\omega^\beta], \]
\[ D\omega^\alpha=[\omega^\beta\omega_\beta^\alpha] +A_{k\beta}^{\alpha}[\omega^k\omega^\beta] +[\lambda_k\omega^k\omega^\alpha]. \tag{4} \]
It is clear from this that the system of equations \(\omega^\alpha=0\) is completely integrable, since the exterior differentials of the forms \(\omega^\alpha\) vanish as a consequence of the system itself. As for the system \(\omega^i=0\), it is completely integrable if and only if all \(A_{k\beta}^{\alpha}\) are equal to zero. This is satisfied, according to the lemma, if the plane \(E_{n-q}\) is odd-dimensional. If, however, \(n-q\) is even, then we additionally assume that \(A_{k\beta}^{\alpha}=0\). This is equivalent to the requirement that the group \(H_1\) admit no rotation commuting with it.
From the complete integrability of the systems \(\omega^\alpha=0\) and \(\omega^i=0\), it follows that the planes \(E_q\) and \(E_{n-q}\) are holonomic, i.e., they envelope the surfaces \(V_q\) and \(V_{n-q}\). Let us pass to a new coordinate system determined by the orthogonal decomposition of \(V_n\) into these surfaces, and denote by the symbol \(d\) differentiation with respect to \(x^i\), and by \(\delta\) differentiation with respect to \(x^\alpha\). Then, first,
\[ \omega^i=\omega^i(d),\qquad \omega^\alpha=\omega^\alpha(\delta), \]
and, secondly, from the structure equations (4), taking into account \(A_{k\beta}^{\alpha}=0\):
\[ \delta \sum_i(\omega^i(d))^2=0, \]
\[ d\sum_\alpha(\omega^\alpha(\delta))^2 = -2\lambda_k\omega^k(d)\sum_\alpha(\omega^\alpha(\delta))^2, \tag{5} \]
\[ \delta\bigl(\lambda_k\omega^k(d)\bigr)=0. \]
To prove the last equality, the equations d) from (2) were also used, from which it follows that \(\overset{0}{L}{}^{\,i}_{j}\lambda_i=0\), i.e., the \(\lambda_i\) determine one-dimensional invariant directions with respect to \(H_0\) on the plane \(E_q\).
It follows immediately from (5) that the metric \(ds^2\) in the indicated coordinate system has the form
\[ ds^2=\sum_i(\omega^i)^2+\sum_\alpha(\omega^\alpha)^2 = g_{ij}(x^k)\,dx^i dx^j+\sigma(x^k)\,a_{\alpha\beta}(x^\gamma)\,dx^\alpha dx^\beta, \tag{6} \]
and, consequently, the space \(V_n\) is semireducible \((^2)\). The group \(G_r\) is an unmixed group \((^3)\) with respect to the decomposition (6); otherwise we would arrive at a contradiction with the assumption of irreducibility of \(H_1\).
Thus, the following theorem has been proved:
Theorem 2. Let there be given a homogeneous Riemannian space \(V_n\) \((ds^2>0)\) with group \(G_r\) and stationary subgroup \(H\). Suppose that the group \(H\) decomposes into the direct product of two subgroups, one of which is irreducible, and if the plane on which it acts in the tangent space of the given point is four-dimensional, it has no mutually permutable rotations. Then the space \(V_n\) is semireducible, and its metric is reduced to the form (6), with respect to which the group \(G_r\) is unmixed.
From Theorem 2 there follows a more general theorem:
Theorem 3. If in a homogeneous proper Riemannian space \(V_n\) the stationary group
\[ H=H_0\times H_1\times\ldots\times H_p, \]
and all groups \(H_t\) \((t>0)\) possess the property indicated in Theorem 2, then the space \(V_n\) is \(\rho\)-fold semireducible, i.e., in some coordinate system
\[ ds^2=ds_0^2(x^i)+\sigma_1(x^i)\,ds_1^2(x^{\alpha_1})+\ldots+\sigma_p(x^i)\,ds_p^2(x^{\alpha_p}). \tag{7} \]
With respect to the decomposition (7), the group \(G_r\) is unmixed.
Remark 1. The function \(\sigma\) in (6) is determined from the condition
\[ \lambda_k\omega^k(d)=-\tfrac{1}{2}\,d\ln\sigma. \]
Therefore, if \(H_0\) does not fix one-dimensional directions, then all \(\lambda_k=0\), i.e., \(\sigma=\mathrm{const}\), and the space \(V_n\) is reducible in accordance with Theorem 1.
Remark 2. Since the transitive group \(G_r\) in (6) is an unmixed group of motions, under motions from \(G_r\) the function \(\sigma\) passes into \(K\sigma\), and \(a_{\alpha\beta}dx^\alpha dx^\beta\) into \(\dfrac{1}{K}a_{\alpha\beta}dx^\alpha dx^\beta\). This means that the hypersurfaces \(\sigma=\mathrm{const}\) form in \(V_q\) a system of imprimitivity for the group of motions induced by \(G_r\) in \(V_q\), while the metric \(a_{\alpha\beta}dx^\alpha dx^\beta\) must be Euclidean.
Remark 3. The requirement that the group \(H_1\) have no mutually permutable rotations is essential, for otherwise, as we have seen, the corresponding planes \(E_{n-q}\) may turn out to be nonholonomic, and there will no longer be a semireducible fibration of the space \(V_n\). For example, the space \(V_4\)
\[ ds^2=e^{x^4}\bigl(dx^{1\,2}+dx^{2\,2}\bigr)+e^{2x^4}\bigl(dx^3+x^1dx^2-x^2dx^1\bigr)^2+dx^{4\,2} \]
admits a transitive group of motions with translation operators
\[ X_1=p_3,\quad X_2=p_2+x^1p_3,\quad X_3=p_1-x^2p_3, \]
\[ X_4=-\frac{1}{2}x^1p_1-\frac{1}{2}x^2p_2-x^3p_3+p_4 \]
and with the rotation operator \(Y=x^1p_2-x^2p_1\) relative to the point \((0,0,0,0)\). Here \(H_0\) is the identity transformation in the plane \(\{x^3,x^4\}\); \(H_1\) is the group of rotations in the Euclidean plane \(\{x^1,x^2\}\). Obviously, \(H_1\) is irreducible, but there is no semireducible decomposition of \(V_4\), in view of the fact that \(H_1\) admits rotations permutable with it—in the present case it is permutable with itself. Analogous examples could be given for any \(n\) (for \(n=5\), see \((4)\)).
Moscow State University
named after M. V. Lomonosov
Received
13 II 1958
REFERENCES
- H. Wakakuwa, Tôhoku Math. J., (2), 6, 121 (1954).
- G. I. Kruchkovich, DAN, 115, No. 5, 862 (1957).
- G. I. Kruchkovich, Uspekhi Mat. Nauk, 12, 6 (78) (1957).
- C. Teleman, Studii şi cerc. mat., 4, No. 3–4, 504 (1953).