Abstract
Full Text
MATHEMATICS
Ya. Shapiro, T. Temirov
On the Question of Riemannian Spaces with Reducible Isotropy Group
(Presented by Academician A. N. Kolmogorov on 27 II 1964)
1. Let a continuous group (G_r) of motions act in a proper Riemannian space (V_n), with a nontrivial reducible isotropy group (H) (for all points of some domain). As is easy to see, one may always assume that the group (H), acting in a proper Euclidean space, has no one-dimensional invariant directions in some invariant plane (E_{n-m}). The plane (E_m) orthogonal to (E_{n-m}) is also invariant with respect to (H), and thus (H = H_0 \times H_1), where (H_0) and (H_1) denote the groups acting on the planes (E_m) and (E_{n-m}).
Choose a (generally speaking, nonholonomic) orthonormal frame (R) so that its first (m) vectors (e_\alpha)* belong to (E_m), and the last (n-m) vectors (e_a) to (E_{n-m}).
Every motion (\varphi \in G_r) induces (at each point of (V_n)) an orthonormal frame (R^), whose vectors (e^{\alpha}, e^*).}) also belong to the planes (E_m, E_{n-m
Thus, if (p) is the transition matrix from (R) to (R^): (R^ = pR), then
[
p^a_{\alpha}=p^\alpha_a=0.
\tag{1}
]
Let (\omega^i, \omega^i_j) be the forms defining the connection of (V_n) with respect to (R); then for the quantities (\omega^{i}, \omega^{i}_j), playing the analogous role with respect to (R^*), one has
[
\omega^{*i}=q^i_j\omega^j,
\tag{2a}
]
[
\omega^{*i}_j=q^i_s\omega^s_k p^k_j+q^i_s dp^s_j,
\tag{2b}
]
where (q^i_s) is the matrix inverse to (p^i_s).
If (x^{\prime i}) are the coordinates of the image of the point (M(x^k)) under the motion (\varphi), then, evidently,
[
\omega^{i}(x', dx')=\omega^i(x, dx),
]
[
\omega^{i}_j(x', dx')=\omega^i_j(x, dx).
\tag{I}
]
Now suppose that (\varphi) belongs to the stationary subgroup of the point (M(x)). If, moreover, it is assumed that the coordinates (x) entering into (I) refer precisely to this point, then (I), with the aid of (2a) and (2b), can be rewritten in the form
[
q^i_j\omega^j(dx')=\omega^i(dx),
\tag{3a}
]
[
\omega^i_k(dx')p^k_j+dp^i_j=p^i_k\omega^k_j(dx).
\tag{3b}
]
Putting (i=a,\ j=\beta), we obtain, in connection with (1),
[
\omega^a_{\sigma}(dx')p^\sigma_{\beta}=p^a_b\omega^b_{\beta}(dx).
\tag{4}
]
* Indices throughout range over the following limits: (\alpha,\beta,\gamma,\ldots=1,\ldots,m;\quad a,b,c,\ldots=m+1,\ldots,n;\quad i,j,k=1,\ldots,n.)
Representing (\omega_j^i) in the form
[
\omega_j^i=\gamma_{jk}^i\omega^k,
]
where (\gamma_{jk}^i) are functions of the coordinates, and replacing in (4), with the aid of (3a), the quantities (\omega^i(dx')) by their expressions in terms of (\omega^k(dx)), we obtain, after eliminating the arbitrary (\omega^k(dx)),
[
\gamma_{\sigma k}^{a}p_i^k p_\beta^\sigma=p_b^a\gamma_{\beta i}^{b},
]
which is equivalent to two groups of equalities
[
\gamma_{\sigma b}^{a}p_c^b p_\beta^\sigma=p_b^a\gamma_{\beta c}^{b},
\tag{5a}
]
[
\gamma_{\sigma\tau}^{a}p_\alpha^\tau p_\beta^\sigma=p_b^a\gamma_{\beta\alpha}^{b}.
\tag{5b}
]
II. Case A. Suppose first that (H_0) (as well as (H_1)) admits no invariant one-dimensional directions. Then from (5a) and (5b) we immediately obtain
[
\gamma_{\sigma k}^{a}=0
\quad\text{or}\quad
\omega_\sigma^a=0.
\tag{6}
]
Case B. Now, without imposing any restrictions on (H_0), suppose that the group (H_1) (having no invariant one-dimensional directions) admits no nontrivial commutators. Then, putting in (5a), (5b) (p_\beta^\sigma=\delta_\beta^\sigma), we obtain
[
\gamma_{\sigma b}^{a}=T_\sigma\delta_b^a,\qquad
\gamma_{\sigma\beta}^{a}=0,
]
where (T_\sigma) are certain quantities, and (\delta_b^a) is the Kronecker delta, or
[
\omega_\sigma^a=T_\sigma\omega^a.
\tag{7}
]
We note that (6) is obtained from (7) when (T_\sigma=0).
From the conditions that the torsion be zero,
[
d\omega^i+\omega_j^i\wedge\omega^j=0,
]
represented in the form of two groups of equalities
[
d\omega^a+\omega_\sigma^a\wedge\omega^\sigma+\omega_b^a\wedge\omega^b=0,
\tag{8a}
]
[
d\omega^\alpha+\omega_b^\alpha\wedge\omega^b+\omega_\beta^\alpha\wedge\omega^\beta=0,
\tag{8b}
]
it now follows, in view of (7) and (\omega_a^\sigma=-\omega_\sigma^a), that each of the systems of Pfaffian equations (\omega^a=0), (\omega^\alpha=0) is completely integrable. Consequently, it is possible to choose local coordinates (x^i) for which (\omega^a) are linear combinations of the differentials (dx^b), while (\omega^\alpha) are differentials (dx^\beta).
Introducing the symbols (d,\delta) of differentiation with respect to the variables (x^\alpha,x^a), respectively, which have the property
[
d\delta=\delta d=0,
]
it is easy from (8a), (8b), taking (7) into account, to obtain
[
\delta\sum_a(\omega^a)^2=0,
\tag{9a}
]
[
d\sum_a(\omega^a)^2=2T_\alpha\omega^\alpha\sum_a(\omega^a)^2.
\tag{9b}
]
In case A the metric
[
ds^2=\sum_\alpha(\omega^\alpha)^2+\sum_a(\omega^a)^2,
]
computed for a special system of local coordinates, has, conse-
therefore, the form
[
ds^2 = g_{\alpha\beta}(x^\sigma)\,dx^\alpha dx^\beta + g_{ab}(x^c)\,dx^a dx^b
]
holds, and the following theorem is valid.
Theorem 1. If the isotropy group (H) of a proper Riemannian space (V_n) is, for all points of some domain, the direct product of two groups (H_0) and (H_1), acting respectively in the orthogonal (G_r)-invariant planes (E_m) and (E_{n-m}) and admitting no one-dimensional fixed directions, then (V_n) is reducible.
Under the assumption that the group (G_r) (for which (H) is the isotropy group) is transitive, this theorem was proved by Wakakuwa ((^1)).
In case B, as follows from (9a) and (9b), (ds^2) has the form
[
ds^2 = g_{\alpha\beta}(x^\sigma)\,dx^\alpha dx^\beta
+ e^{2\theta(x^i)}\Pi_{ab}(x^c)\,dx^a dx^b,
\tag{10}
]
where (\theta) is some function of all the coordinates.
Starting from the fact that the curvature tensor of (V_n) is preserved by the isotropy group, one can prove that (under a certain normalization) (\theta) depends only on the coordinates (x^\sigma).
The metric (10) (with (\theta=\theta(x^\sigma))) characterizes a (V_n) containing a geodesic field of directions ((^2)); it has been considered by many authors and is known under the name semi-reducible ((^3)).
In connection with the above, the following theorem may be formulated.
Theorem 2. If the isotropy group (H) of a proper Riemannian space is the direct product of groups (H_0) and (H_1), acting respectively in the (G_r)-invariant orthogonal planes (E_m) and (E_{n-m}), with (H_1) admitting no invariant one-dimensional directions nor nontrivial commutators, then the metric of (V_n) is semi-reducible.
Under the assumption that the group (G_r) is transitive, the main content of this theorem coincides with the theorem proved by G. I. Kruchkovich and Gu Chao-hao ((^4)).
As noted in their work, the metric (\Pi_{ab}dx^a dx^b) is (in the case of transitivity of (G_r)) Euclidean. If, however, transitivity of (G_r) is not required, then for any metric (\Pi_{ab}dx^a dx^b) with an isotropy group admitting no one-dimensional invariant directions and no nontrivial commutators, the space with metric (10) admits a group of motions for which the conditions formulated in Theorem 2 hold.
Gorky State University
named after N. I. Lobachevsky
Received
14 XI 1963
CITED LITERATURE
- H. Wakakuwa, Tôhoku Math. J., 2, 6, 121 (1954).
- Ya. L. Shapiro, DAN, 32, No. 4, 237 (1941).
- G. I. Kruchkovich, DAN, 115, No. 5 (1957).
- G. I. Kruchkovich, Gu Chao-hao, DAN, 120, No. 6, 171 (1958).