Abstract
Full Text
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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