MATHEMATICS

GU CHAO-HAO

ON CERTAIN TYPES OF HOMOGENEOUS RIEMANNIAN SPACES

(Presented by Academician P. S. Aleksandrov on 29 IV 1958)

1. Let (V_n) be a homogeneous Riemannian space with positive-definite metric, (G) its group of motions, and (H) the stationary group of a given point (P_0). If (E) is the tangent space at this point, then the group (H) may be regarded as a group of rotations (the isotropy group) in (E).

In the work of G. I. Kruchkovich and the author ({}^{1}) the following theorem was obtained:
If the isotropy group (H) decomposes into the direct product of two subgroups (H = H_0 \times H_1), where (H_1) is irreducible and admits no rotations permutable with it on the plane (E_1) on which it acts in (E), while (H_0) acts on the orthogonal complement to (E_1), then the space (V_n) is semi-reducible, i.e. its metric can be reduced to the form

[
ds^2 = g_{i_0 j_0}(x^{k_0})\, dx^{i_0} dx^{j_0}
+ \sigma(x^{k_0}) g_{i_1 j_1}(x^{k_1})\, dx^{i_1} dx^{j_1}
\tag{1}
]

[
(i_0, j_0 = 1, \ldots, q,\qquad i_1, j_1 = q+1, \ldots, n)
]

and the group (G) is a non-mixing group of this space ({}^{2}).

The purpose of the present note is to study such homogeneous Riemannian spaces (V_n) for which the isotropy group (H) acts irreducibly on the plane (E_1) and admits there rotations permutable with it. It is assumed here that the orthogonal complement (E_0) to (E_1) in the tangent space (E) consists entirely of fixed directions of the group (H).

1. Let ({M, e_1, \ldots, e_n}) be a system of orthonormal frames in (V_n), admissible relative to the group of motions (G), and

[
dM = \omega^i e_i,\qquad de_i = \omega_i^j e_j \quad (i,j = 1,\ldots,n)
\tag{2}
]

the equations of infinitesimal displacements of these frames; (\omega^i, \omega_i^j) are invariant forms of the group (G). The Maurer–Cartan equations hold:

[
d\omega^i = \frac{1}{2} C^i_{jk}[\omega^j \omega^k] + a^i_{j\rho}[\omega^j \theta^\rho]
\quad (\rho, r = n+1,\ldots,r);
\tag{3}
]

[
d\theta^\rho =
\frac{1}{2} C^\rho_{\lambda\mu}[\theta^\lambda \theta^\mu]
+ C^\rho_{l\tau}[\omega^l \theta^\tau]
+ \frac{1}{2} C^\rho_{lm}[\omega^l \omega^m],
\tag{4}
]

where (\omega^i, \theta^\rho) are independent forms among (\omega^i, \omega_i^j); (C^i_{jk}, C^\rho_{\lambda\mu}, C^\rho_{l\tau}, a^i_{j\rho}, C^\rho_{lm}) are the structural constants of the group (G). They satisfy the relations following from the Jacobi identities:

[
a^i_{l\rho} a^l_{j\sigma} - a^i_{l\sigma} a^l_{j\rho}
= C^\tau_{\sigma\rho} a^i_{j\tau},
\tag{5}
]

[
a^k_{l\tau} C^\tau_{mp} - a^k_{m\tau} C^\tau_{l\rho}
= C^i_{lm} a^k_{i\rho} + C^k_{il} a^i_{m\rho} - C^k_{im} a^i_{l\rho},
\tag{6}
]

[
a^k_{l\tau} C^\tau_{mn} + a^k_{m\tau} C^\tau_{nl} + a^k_{n\tau} C^\tau_{lm}
= C^i_{lm} C^k_{in} + C^i_{mn} C^k_{il} + C^i_{nl} C^k_{im}.
\tag{7}
]

and so on. Here (A_\rho=|a^i_{j\rho}|) are the matrices of infinitesimal rotations of the isotropy group.

If the frames are chosen so that (e_1,\ldots,e_q) lie in (E_0), and (e_{q+1},\ldots,e_n) in (E_1), then

[
a^{i_0}{j\rho}=a^j=0.
\tag{8}
]

As a consequence of this, from equations (6) we obtain the following results:

1) (C^{i_1}{j_1k_0}=0,\quad C^{i_0}=0.)

2) The matrices (C^{i_0}=|C^{i_0}{i_1k_1}|) commute with the matrices (A\rho) for all (i_0) and (\rho).

3) Each of the operators
[
X_{k_0}=C^{i_1}_{j_1k_0}x^{j_1}\frac{\partial}{\partial x^{i_1}}
]
generates a linear normalizer of the group (H) in (E_1).

1. In what follows the following algebraic propositions will be used.

Lemma 1. Let (H) be an irreducible orthogonal group acting in the (m)-dimensional space (E_m); let (H') be the set of all rotations in (E_m) commuting with each rotation from (H), which evidently also forms a group. Then only three cases are possible: 1) (H') contains no rotation; 2) (H') is a one-parameter group; 3) (H') is a three-parameter group (H'_3).

Lemma 2 ((^3)). If there exists a rotation (g) commuting with the group (H), then (m=2s), and in a suitable coordinate system the rotation (g) is represented by the matrix

[
I_1=\left|\begin{array}{cc}
0 & E_s\
-E_s & 0
\end{array}\right|
\quad (E_s\text{ is the identity matrix of order }s)
\tag{9}
]

in the orthogonal algebra (D_s).

Lemma 3. If (H'=H'3), then (m=4s), and in a special coordinate system (H'_3) corresponds to a subalgebra of the orthogonal algebra (D) with basis matrices

[
I_1=\left|\begin{array}{cc}
0 & E_{2s}\
-E_{2s} & 0
\end{array}\right|,\qquad
I_2=\left|\begin{array}{cc}
A & B\
B & -A
\end{array}\right|,\qquad
I_3=\left|\begin{array}{cc}
B & -A\
-A & -B
\end{array}\right|,
\tag{10}
]

where (A,B) are skew-symmetric commuting matrices of order (2s) satisfying the equality
[
A^2+B^2=-E_{2s}.
]
Let us note that (I_1,I_2,I_3) and (E_{4s}) constitute a basis of the matrix representation of the field of quaternions.

Lemma 4. If (g) is a one-dimensional linear normalizer of an irreducible orthogonal group (H) in (E_m), then (g) is represented by the matrix (\lambda E+A+B), where (E) is the identity matrix, (A=|a^\alpha_\beta|) belongs to the Lie algebra of the group (H), and (B=|b^\alpha_\beta|) belongs to the Lie algebra of the group (H').

1. If the isotropy group (H) admits no rotations commuting with it, then, according to the theorem cited above, the space (V_n) is semireducible with metric (1). In the case (\sigma\ne\mathrm{const}), this theorem can be somewhat sharpened; namely, (g_{i_1j_1}(x^{k_1})\,dx^{i_1}dx^{j_1}) in (1) must be Euclidean. Therefore one can choose the coordinate system so that

[
ds^2=g_{i_0j_0}(x^{k_0})\,dx^{i_0}dx^{j_0}
+e^{-2x^1}\bigl[(dx^{q+1})^2+\cdots+(dx^n)^2\bigr].
\tag{11}
]

Moreover, we also obtain the following theorem.

Theorem 1. A homogeneous Riemannian space (V_n) ((ds^2>0)) does not admit a group of similarities broader than the group of motions, except in the case when (V_n) is Euclidean.

1. Let us now consider the case when the isotropy group (H) admits only a one-parameter group of permutable rotations. Then equations (2) can be written in the form

[
d\omega^1=\frac12 C^1_{j_0 k_0}[\omega^{j_0}\omega^{k_0}]
+ C\sum_\alpha [\omega^\alpha,\omega^{\alpha+s}]
\qquad
\left(s=\frac{n-q}{2},\ \alpha=q+1,\ldots,q+s\right);
]

[
d\omega^a=\frac12 C^a_{j_0 k_0}[\omega^{j_0}\omega^{k_0}]
\qquad (a=2,\ldots,q),
\tag{12}
]

[
d\omega^{i_1}
=\frac12 C^{i_1}{j_1 k_1}[\omega^{j_1}\omega^{k_1}]
+ C]}[\omega^{i_1}\omega^{m_0
+ b^{i_1}{l_1 m_0}[\omega^{l_1}\omega^{m_0}]
+ d^{i_1}\theta^\rho],}[\omega^{j_1
]

where the matrices (B_{m_0}=|b^{i_1}{l_1m_0}|) are permutable with the matrices (A\rho) for all (m_0) and (\rho); (C) is a constant.

a) If (C=0), then we return to case (11), or to the reducible metric (1) with (\sigma=\mathrm{const}).

b) If (C\ne 0), and all (C_{m_0}=0), then in some coordinate system the metric of the space is reduced to the form

[
ds^2=ds_0^2(x^a)+
\left[dx^1+\omega(x^a,dx^a)+\frac{C}{2}\sum_\alpha
(x^\alpha dx^{s+\alpha}-x^{s+\alpha}dx^\alpha)\right]^2
+ds_1^2(x^{i_1}),
\tag{13}
]

where (ds_0^2(x^a)) is a ((q-1))-dimensional metric admitting a simply transitive group of motions (G^{(1)}); the forms
(\overline{\omega}^1=dx^1+\omega(x^a,dx^a)\omega^a) are invariant forms of the group of motions (G_q), consisting of (G^{(1)}) and a one-dimensional center; (ds_1^2(x^{i_1})) is an ((n-q))-dimensional metric with a transitive group of motions (G^{(2)}), leaving the exterior form
(\sum_\alpha[dx^\alpha dx^{\alpha+s}]) invariant.

c) If (C\ne0) and not all (C_{m_0}=0), then

[
ds^2=ds_0^2(x^a)+e^{-2f(x^a)}
\left[dx^1+\omega(x^a,dx^a)+\frac12 C\sum_\alpha
(x^\alpha dx^{\alpha+s}-x^{\alpha+s}dx^\alpha)\right]^2+
]

[
+\,e^{-2f(x^a)}\left[(dx^{q+1})^2+\cdots+(dx^n)^2\right],
\tag{14}
]

where (ds_0^2) is a ((q-1))-dimensional metric admitting a simply transitive group (G^{(1)}); (df=C_a\omega^a); the forms
(\overline{\omega}^1=e^{-2f(x^a)}[dx^1+\omega(x^a,dx^a)]), (\omega^a) are invariant forms of a certain group (G_q), which has a ((q-1))-dimensional normal divisor with a continuous center; (G^{(1)}) is a subgroup of (G_q).

Let us note that in cases b) and c) (x^{i_1}=\mathrm{const}) are totally geodesic and imprimitivity submanifolds; however, the entire metric (ds^2) is not semireducible, since the field of planes (E_1) is nonholonomic. The geodesic lines (x^1=t) form a system of imprimitivity of the group of motions and themselves are trajectories of a screw displacement.

1. We pass to a more complicated case, when (H) admits a three-parameter group of rotations permutable with itself. We denote by (\Omega^1,\Omega^2,\Omega^3) the exterior forms of the second order corresponding to the skew-symmetric matrices (I_1,I_2,I_3). Then equations (3) have the form

[
d\omega^{i_0}=\frac12 C^{i_0}{j_0 k_0}[\omega^{j_0}\omega^{k_0}]
+ C^{i_0}\Omega^p;
\tag{15}
]

[
d\omega^{i_1}
=\frac12 C^{i_1}{j_1 k_1}[\omega^{j_1}\omega^{k_1}]
+ C]}[\omega^{i_1}\omega^{m_0
+ b^{i_1}{j_1m_0}[\omega^{j_1}\omega^{m_0}]
+ d^{i_1}\theta^\rho].}[\omega^{j_1
\tag{16}
]

Here the following cases are possible:

a) All (C^{i_0}_{p}=0); then the results of item 4 hold.

b) All (q) forms (C^{i_0}_{p}\Omega^p) are proportional; then the results of item 5 are obtained.

c) Among the forms (C^{i_0}_{p}\Omega^p), 2 forms are dependent.

d) The forms (C^{i_0}_{p\Omega}\Omega^p) are expressed in terms of 3 independent forms from among them.

Let us first consider case d). In this case the metric of the space has one of the following two forms:

[
ds^2=ds_0^2(x^a)+g_{pt}\omega^p\omega^t+ds_1^2(x^{i_1})
\tag{17}
]

[
(a=4,5,\ldots,q;\quad p,t,u=1,2,3);
]

[
ds^2=ds_0^2(x^a)+g_{pt}\omega^p\omega^t+e^{-2f(x^a)}[(dx^{q+1})^2+\cdots+(dx^n)^2],
\tag{18}
]

where (ds_0^2=\sum_a(\omega^a)^2) is a ((q-3))-dimensional metric admitting a simply transitive group (G^{(1)});

[
\omega^p=\varphi_t^p(x^a)[\bar{\omega}^t(x^u,dx^u)+\pi^t(x^a,dx^a)+\sigma^t(x^{i_1},dx^{i_1})];
\tag{19}
]

(\varphi_u^p(x^a)) are determined from the completely integrable system

[
d\varphi_t^p=-C^p_{ua}\varphi_t^u\omega^a;
\tag{20}
]

(g_{pt}) are constants corresponding to a symmetric positive matrix of order 3. Moreover,

1) (\bar{\omega}^1,\bar{\omega}^2,\bar{\omega}^3) satisfy the relations

[
d\bar{\omega}^1=\alpha[\bar{\omega}^2\bar{\omega}^3],\quad
d\bar{\omega}^2=\beta[\bar{\omega}^3\bar{\omega}^1],\quad
d\bar{\omega}^3=\gamma[\bar{\omega}^1\bar{\omega}^2],
\tag{21}
]

where (\alpha,\beta,\gamma) are constants, equal to zero in case (18);

2) the forms (\varphi_t^p(x^a)[\bar{\omega}^t+\pi^t]), (\omega^a) determine the group (G_q), which contains a ((q-1))-dimensional normal divisor in case (18);

3) the forms (\sigma^t) satisfy the relations

[
\Omega^p=\varphi_t^p d\sigma^t;
\tag{22}
]

4) (ds_1^2(x^{i_1})) in (17) is an ((n-q))-dimensional metric admitting a group of motions with isotropy group (H), which leaves invariant the exterior forms (d\sigma^t).

We note that the space has (\infty^{\,n-3}) imprimitive three-dimensional completely geodesic submanifolds (V_3): (x^a=\mathrm{const}), (x^i=\mathrm{const}), each geodesic line of which is the trajectory of a displacement. In case (18) each (V_3) has a Euclidean metric.

In case c) the results (17) and (18) are preserved, only (p,t=1,2), (a=3,\ldots,q), and (d\bar{\omega}^1=d\bar{\omega}^2=0).

We formulate the results obtained in the following theorem.

Theorem 2. If the isotropy group of a homogeneous Riemannian space (V_n) leaves fixed all directions in the plane (E_0) and acts irreducibly on the orthogonal complement (E_1) to (E_0), then the metric of the space has the form (1) (in it (\sigma(x^{k_0})=1)), (11), (13), (14), (17), or (18).

We note that in cases (11), (14), (18) the metric is determined by specifying the group (G_q) with the properties indicated above, and in cases (1), (13), (17) by the group (G_q) and an ((n-q))-dimensional metric admitting a transitive group of motions with isotropy group (H), which leaves invariant one or three exterior forms of the second order in cases (13) and (17), respectively.

In conclusion I express my sincere gratitude to P. K. Rashevskii, A. M. Vasil’ev, and G. I. Kruchkovich for valuable assistance.

Moscow State University
named after M. V. Lomonosov

Received
24 IV 1958

References

1. G. I. Kruchkovich, Gu Chao-hao, DAN, 120, No. 6 (1958).
2. G. I. Kruchkovich, Uspekhi Mat. Nauk, 12, 6(78), 149 (1957).
3. É. Cartan, Geometry of Lie Groups and Symmetric Spaces, IL, 1949, p. 162.