MATHEMATICS

G. I. KRUCHKOVICH

ON SEMI-REDUCIBLE RIEMANNIAN SPACES

(Presented by Academician P. S. Aleksandrov, 16 III 1957)

1°. We shall call a Riemannian space \(V_n\) semi-reducible if there exists a coordinate system in which its metric form has the form

\[ ds^2 = ds_0^2 + \sigma ds_1^2 = g_{ij}(x^k)\,dx^i dx^j + \sigma(x^k)\,a_{\alpha\beta}(x^\gamma)\,dx^\alpha dx^\beta \]
\[ (i,\ j,\ k = 1,\ldots,q;\ \alpha,\ \beta,\ \gamma = q+1,\ldots,n), \tag{1} \]

where \(ds_0^2\) and \(ds_1^2\) are independent \(q\)-dimensional and \((n-q)\)-dimensional metrics, each depending on its own arguments, while the function \(\sigma\) depends only on the variables of \(ds_0^2\). In particular, for \(\sigma=\mathrm{const}\), semi-reducible \(V_n\) include all reducible Riemannian spaces. We shall call the metric \(ds_0^2\) the principal part of \(ds^2\), and \(ds_1^2\) its additional part.

Semi-reducible \(V_n\) encompass a broad class of Riemannian spaces having the most diverse geometric properties. Thus, all spaces of constant curvature and Kagan subprojective spaces are semi-reducible, since they admit a coordinate system in which*

\[ ds^2 = dx^{0\,2} + \sigma(x^0)\,ds_1^2(x^1,\ldots,x^{n-1}), \tag{2} \]

where \(ds_1^2\) is a metric of constant curvature \(K_1\). The \(V_n\) of constant curvature are singled out from (2) by the condition \(\sigma(\ln\sigma)''=-2K_1\). In the study of motions in Riemannian spaces it turned out that all the most mobile \(V_n\), with isolated exceptions, are semi-reducible \((^1,^2)\). Semi-reducible \(V_n\) also appear in connection with conformal \((^3)\) and projective \((^4)\) transformations in \(V_n\). Let us also note that in the theory of relativity the so-called centrally symmetric space-time \((^5)\), whose metric is likewise semi-reducible, is of great importance.

It is natural to pose the question of identifying properties characterizing the entire class of semi-reducible spaces \(V_n\). In the present note a geometric characteristic is indicated and a tensor criterion for semi-reducible \(V_n\) is introduced; the question of the uniqueness of the representation of the metric of a proper Riemannian space in the form (1) is also considered.

2°. From (1) it is clear that a semi-reducible \(V_n\) is foliated into mutually orthogonal \(q\)-dimensional and \((n-q)\)-dimensional surfaces carrying the metrics \(ds_0^2\) and \(\sigma ds_1^2\). Here the first surfaces are totally geodesic, while the second are similar to one another and consist, as is not difficult to verify, of umbilical points. Such a foliation of \(V_n\) turns out also

\[ \text{* Exceptional subprojective spaces are not included here; their metric } ds^2=2dx^1dx^2+\sigma(x^2)(e_3dx^{3\,2}+\cdots+e_ndx^{n\,2}),\ e_\alpha=\pm1, \text{ is also semi-reducible.} \]

and sufficient for its semi-reducibility, namely, the following theorem holds.

Theorem 1. In order that a Riemannian space \(V_n\) be semi-reducible, it is necessary and sufficient that it be foliated into two families of \(q\)-dimensional and \((n-q)\)-dimensional mutually orthogonal surfaces; moreover, the surfaces of one family are totally geodesic, while the surfaces of the other are umbilical and mutually similar.

3°. If in the space (1) one takes a symmetric tensor \(A_{ab}\) with components \(A_{ij}=A_{i\alpha}=0,\ A_{\alpha\beta}=\sigma a_{\alpha\beta}\), then it is easy to see that the equations

\[ A_{ab,c}=-\frac12(u_aA_{bc}+u_bA_{ac}), \tag{3} \]

\[ A_{ac}A^c_b=A_{ab}\qquad (a,b,c=1,\ldots,n), \tag{4} \]

are satisfied, where the comma denotes covariant differentiation in \(V_n\); \(u_a\) is the gradient of the function \(u=\ln\sigma\). At the same time, evidently, \(A_{ab}\ne \lambda g_{ab}\).

Conversely, suppose that for some symmetric tensor \(A_{ab}\ne\lambda g_{ab}\) and for a gradient \(u_a\) equations (3) and (4) are satisfied. It follows from (4) that the matrix \(A=\|A^a_b\|\) has simple elementary divisors. This means that at each point the tensors \(A_{ab}\) and \(g_{ab}\) can be simultaneously reduced to the form \(g_{ab}=\delta_{ab},\ A_{ab}=\rho_a\delta_{ab}\), where \(\rho_a\) are the roots of the equation \(|A_{ab}-\rho g_{ab}|=0\); \(\delta_{ab}\) are the Kronecker symbols. Using equations (3), one can show that the principal planes corresponding to different roots \(\rho_\alpha\) are holonomic; and then, passing to the special coordinate system determined by these planes, it is no longer difficult to show that the space \(V_n\) is semi-reducible.

Theorem 2. In order that a Riemannian space \(V_n\) be semi-reducible, it is necessary and sufficient that there exist a symmetric tensor \(A_{ab}\), not proportional to the metric tensor, which together with some gradient \(u_a\) satisfies the equations:

\[ \text{1) }\ A_{ab,c}=-\frac12(u_aA_{bc}+u_bA_{ac});\qquad \text{2) }\ A_{ac}A^c_b=A_{ab}. \]

This theorem is a direct generalization of the known criterion for reducibility \({}^{(6)}\), which in our case is obtained for \(u_a=0\), i.e. \(\sigma=\mathrm{const}\). We note that for proper Riemannian spaces \(V_n\) \((ds^2>0)\) condition 2) of the theorem is superfluous, in view of the fact that in this case the matrix \(\|A_{ab}-\rho g_{ab}\|\) always has simple elementary divisors. Consequently, in a proper Riemannian space every nontrivial (i.e. \(\ne\lambda g_{ab}\)) symmetric tensor \(A_{ab}\) satisfying equations (3) determines a semi-reducible decomposition of the metric \(ds^2\).

4°. Suppose that a semi-reducible space (1) is given. The question arises whether there exists another foliation of \(V_n\) into surfaces, geometrically distinct from the given one, as indicated in Theorem 1; that is, whether the representation of the metric \(ds^2\) in the form (1) is unique up to trivial transformations of the form \(\tilde{x}^i=f^i(x^k),\ \tilde{x}^{\alpha}=\varphi^\alpha(x^\gamma)\).

We shall show by examples that uniqueness of the representation of the metric in semi-reducible form may fail:

\[ \text{1) }\ ds^2=(ds_0^2+\sigma_1ds_1^2)+\sigma_2ds_2^2 =(ds_0^2+\sigma_2ds_2^2)+\sigma_1ds_1^2, \]
where \(ds_0^2,\ ds_1^2,\ ds_2^2\) are independent metrics; \(\sigma_1,\sigma_2\) depend only on the coordinates of the metric \(ds_0^2\).

\[ \text{2) }\ ds^2=ds_0^2+\sigma ds_1^2, \]
where \(ds_1^2=d\tau_0^2+\nu d\tau_1^2\); then
\[ ds^2=(ds_0^2+\sigma d\tau_0^2)+\sigma\nu d\tau_1^2. \]

\[ \text{3) }\ ds^2=ds_0^2+\sigma ds_1^2, \]
where
\[ ds_0^2=d\tau_0^2(x^1,\ldots,x^p)+\lambda(x^1,\ldots,x^p)d\tau_1^2(x^{p+1},\ldots,x^q), \]
\[ \sigma=\lambda(x^1,\ldots,x^p)\nu(x^{p+1},\ldots,x^q); \]
then
\[ ds^2=d\tau_0^2+\lambda(d\tau_1^2+\nu ds_1^2). \]

In the first case, several semi-reducible decompositions of the metric \(ds^2\) appeared because, for one principal part \(ds_0^2\), it contained several additional metrics \(ds_\alpha^2\). In the second example the principal part \(ds_0^2\) increased owing to the semi-reducible decomposition of the additional metric \(ds_1^2\). We shall say that in this case the new semi-reducible decomposition of the metric \(ds^2\) is a continuation to the right of the old one. Finally, in the third example we are dealing with a continuation to the left of the metric \(ds^2\), when the additional part increased owing to the semi-reducibility of \(ds_0^2\).

If one restricts oneself to proper Riemannian spaces, then it turns out that, for inhomogeneous \(ds_0^2\), there can be no other, more complicated examples of violation of uniqueness of the semi-reducible decomposition of the metric \(ds^2\). More precisely, the following theorem holds:

Theorem 3. If the metric of a proper Riemannian space has, in some coordinate system, the form

\[ ds^2=ds_0^2+\sigma_1(x^i)ds_1^2+\cdots+\sigma_p(x^i)ds_p^2, \tag{5} \]

where \(ds_0^2\) is an inhomogeneous and irreducible metric, and the functions \(\sigma_1,\ldots,\sigma_p\) are not pairwise in a constant ratio, then the corresponding foliation of the space \(V_n\) is unique, i.e. every other representation of \(ds^2\) in the form (5) is obtained from the given one by trivial transformations.*

Thus, different semi-reducible decompositions of \(ds^2\) of the form (1) are obtained from (5) either by choosing the additional metric \(ds_\alpha^2\) and assigning all the rest to the principal part, or by continuation to the right, if some metric \(ds_\alpha^2\) is itself semi-reducible.

If in (1) \(ds_0^2\) is semi-reducible, then the metric \(ds^2\) must be continued as far as possible to the left, and if the latter principal part proves to be inhomogeneous, then Theorem 3 can be applied, after first separating from the principal part all the remaining additional metrics \(ds_\alpha^2\), if any exist.

\(5^\circ\). Let us next consider the case of a semi-reducible proper Riemannian space \(V_n\) with one-dimensional principal part

\[ ds^2=dx^{0\,2}+\lambda(x^0)ds_1^2(x^1,\ldots,x^{n-1}). \tag{6} \]

In this case Theorem 3 is not true. For example, on the Euclidean plane one can choose, in an infinite number of ways, a polar coordinate system in which \(ds^2=d\rho^2+\rho^2d\varphi^2\).

Lemma (cf. \((3)\)). In order that \(V_n\) be semi-reducible with one-dimensional principal part, it is necessary and sufficient that there exist a solution \(f\ne\mathrm{const}\), \(\varphi\) of the equations **

\[ f_{,ab}=\varphi g_{ab}. \tag{7} \]

Here it is necessary that \(\varphi=\varphi(f)\).

Different representations of the given metric \(ds^2\) in the form (6) are determined by different solutions of equations (7). Therefore, from all possible representations of the metric under consideration in semi-reducible form, there can

* I.e. transformations of coordinates inside each metric \(ds_0^2,\ldots,ds_p^2\), a renumbering of the metrics, or replacement of \(\sigma_\alpha\) by \(c\sigma_\alpha\) and of \(ds_\alpha^2\) by \(\frac{1}{c}ds_\alpha^2\).

** For an indefinite metric, when \(\varphi=0\), one must also require non-isotropy of the gradient \(f_{,\alpha}\).

one canonical form may be chosen, namely the one in which \(ds_0^2\) is determined by all functionally independent solutions of equations (7). If there is more than one such solution, then \(ds_0^2\) has constant curvature \(K\), and the function \(\sigma\) is such that \(ds^{*2}=ds_0^2+\sigma dt^2\) has the same constant curvature \(K\). Such spaces \(V_n\), in accordance with \((4)\), will be denoted by \(V_0(K)\).

Theorem 4. If a semireducible \(V_n\) (1) admits nontrivial solutions of equations (7), then either it is a space \(V_0(K)\), or every representation (1) of it is obtained from (6) by a right-hand continuation.

Moscow Power Engineering Institute

Received
15 III 1957

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