MATHEMATICS

A. M. VASIL'EV

INVARIANT AFFINE CONNECTIONS IN HOMOGENEOUS SPACES

(Presented by Academician P. S. Aleksandrov on 27 XI 1959)

In the present paper a class, more general than in paper ((^1)), of invariant affine connections in homogeneous spaces with motion group (G) and stationary subgroup (g) is constructed and studied, under the assumption that (g) is equipped and is not a maximal subgroup of the group (G).

1. In this section and in Sec. 2 we give a number of definitions and results necessary for understanding the paper (see also ((^1))).

A subgroup (g) of the Lie group (G) is called equipped if it contains no nontrivial normal divisor of the group (G) and if in the Lie algebra (\hat G) of the group (G) a subspace (H) is singled out, invariant with respect to the subgroup (g^*) of the adjoint group corresponding to the subgroup (g) of the group (G), and such that (\hat g \dot{+} H = \hat G,\ \hat g \cap H = 0), where (\hat g) is the subalgebra corresponding to (g). If the basic left-invariant forms of the group (G), (\omega^a, \omega^i), are chosen so that the equations (\omega^a=0) define (\hat g), and the equations (\omega^i=0) the subspace (H), then the structure equations take the form (cf. ((^2)))

[
D\omega^a=C^a_{ib}[\omega^i\omega^b]+\frac12 C^a_{db}[\omega^d\omega^b],
]

[
D\omega^i=\frac12 C^i_{kl}[\omega^k\omega^l]+\frac12 C^i_{db}[\omega^d\omega^b].
]

We denote by (d_t\omega) the expression (dt\cdot d(\omega/dt)). The equations

[
d_t\omega^a=C^a_{ib}\omega^i\omega^b+\frac12\gamma^a_{db}\omega^d\omega^b
\tag{1}
]

define the canonically parametrized geodesic lines of a completely determined affine connection of the homogeneous space (G/g). The connection will be invariant if (\gamma^a_{db}=\mathrm{const}) and

[
\gamma^a_{fb}C^f_{id}+\gamma^a_{df}C^f_{ib}-\gamma^f_{db}C^a_{if}=0.
]

The equations

[
d_t\omega^a=\xi C^a_{ib}\omega^i\omega^b,\qquad d_t\omega^i=0
\tag{2}
]

define, for (\xi=\mathrm{const}), in the group (G) the geodesic lines of a certain affine connection without torsion (\Gamma_\xi), invariant with respect to the group (G_l\times g_{\mathrm{pr}}), i.e. all left translations of the group (G) and all its right translations determined by elements of the subgroup (g). These geodesics are the trajectories of the group (G_l\times g_{\mathrm{pr}}).

Let a subgroup (g_0) of the group (G) be contained in (g) and be equipped, and let its equipping space (H_0) contain (H). Then to every connection (\Gamma_\xi) with geodesics (2) there corresponds a certain invariant connection (\Gamma'\xi) in the space (G/g_0). If (g_0) is the stationary subgroup of a point (M) of this space, then the geodesic connections (\Gamma'\xi) passing through (M),

are natural projections of the geodesics of the connection (\Gamma_\xi) tangent to the subspace (H_0).

1. Let there be given in the Lie group (G) a sequence of (n) subgroups (g_{(\lambda)}), endowed with subspaces (H_\lambda), where
   [
   G \supset g_{(1)} \supset \cdots \supset g_{(n)}
   ]
   and
   [
   H_1 \subset \cdots \subset H_n \subset \widehat G .
   ]
   Choosing left-invariant forms (\omega^a, \omega^{i\lambda}) ((\lambda=1,\ldots,n)) in such a way that the equations (\omega^a=0,\omega^{i\nu}=0) ((\nu<\lambda)) define the subgroup (g_{(\lambda)}), and the equations (\omega^{i\mu}=0) ((\mu\geq\lambda)) define the subspace (H_\lambda), we obtain the structural equations in the form
   [
   D\omega^a=C^a_{i_\lambda b}[\omega^{i\lambda}\omega^b]+\frac12 C^a_{db}[\omega^d\omega^b],
   ]
   [
   D\omega^{i\lambda}
   =
   C^{i\lambda}{k\mu l_\lambda}[\omega^{k\mu}\omega^{l\lambda}]
   +\frac12 C^{i\lambda}{k\nu l_\nu}[\omega^{k\nu}\omega^{l\nu}]
   +\frac12 C^{i\lambda}{db}[\omega^d\omega^b],
   ]
   where (\lambda=1,\ldots,n;\ \mu>\lambda;\ \nu\leq\lambda). The equations
   [
   d_t\omega^a=\xi\lambda C^a_{i_\lambda b}\omega^{i\lambda}\omega^b,
   ]
   [
   \tag{3}
   d_t\omega^{i\lambda}=\xi_{\mu\lambda}C^{i\lambda}{k\mu l_\lambda}\omega^{k\mu}\omega^{l\lambda}
   ]
   define on the group (G), for arbitrary constants (\xi_\lambda,\xi_{\mu\lambda}), the geodesic lines of a certain torsion-free affine connection invariant with respect to the group (G_\ell\times g_{(n)\mathrm{pr}}). In (1) a geometric characteristic is given for the geodesic lines of a certain (n)-parameter family of connections of this type. In the present work the same is done for arbitrary connections with geodesics (3).

2. In this paragraph we set forth the basic construction leading to the desired results.

Consider the space of the group (G) as a homogeneous space with the motion group (G_\ell\times g_{(1)\mathrm{pr}}). The stationary subgroup of the identity point will be the group (\bar g_{(1)}) of all inner automorphisms generated by elements of the subgroup (g_{(1)}). The invariant forms (\theta^{i\lambda}) of the group (g_{(1)\mathrm{pr}}) may be chosen so that (g_{(1)}) is defined by the equations (\omega^a=0,\ \bar\omega^{i\lambda}=0), where
[
\bar\omega^{i\lambda}=\omega^{i\lambda}-\theta^{i\lambda}.
]

Consider in the space of the group (G_\ell\times g_{(1)\mathrm{pr}}):

a) the Pfaff system
[
\omega^{i\lambda}=h_\lambda\bar\omega^{i\lambda};
]

b) the system of differential equations
[
d_t\omega^a=\gamma^a_{i_\lambda b}\omega^{i\lambda}\omega^b+\frac12\gamma^a_{db}\omega^d\omega^b,
\tag{b(1)}
]
[
d_t\omega^{i\lambda}=\gamma^{i\lambda},}\omega^{k\mu}\omega^{l\lambda
\tag{b(2)}
]
[
d_t\theta^{i\lambda}=\delta^{i\lambda};}\theta^{k\mu}\theta^{l\lambda
\tag{b(_3)}
]

c) the system of equations
[
d_t\omega^a
=
C^a_{i_\lambda b}\theta^{i\lambda}\omega^b
+
\Gamma^a_{i_\lambda b}\bar\omega^{i\lambda}\omega^b
+
\frac12\Gamma^a_{db}\omega^d\omega^b,
]
[
d_t\bar\omega^{i\lambda}
=
C^{i\lambda}{k\mu l_\lambda}\bigl(\theta^{k\mu}\bar\omega^{l\lambda}-\theta^{l\lambda}\bar\omega^{k\mu}\bigr)
+
C^{i\lambda}{k\nu l_\nu}\theta^{k\nu}\bar\omega^{l\nu}
+
\Gamma^{i\lambda}{k\mu l_\lambda}\bar\omega^{k\mu}\omega^{l\lambda},
]
where the quantities (h,\gamma,\delta,\Gamma) are constants and (\mu>\lambda).

We note that system a) is invariant with respect to all left translations of the group (G_\ell\times g_{(1)\mathrm{pr}}) and with respect to its right translations generated by elements of the subgroup (\bar g_{(n)}) of the group (\bar g_{(1)}), corresponding to the subgroup (g_{(n)}) of the group (G).

We now require that: 1) every integral curve of system b) satisfying system a) at one of its points should satisfy it identically; 2) every parametrized curve satisfying a)

and b), also satisfying a) and c), and conversely. For this it is necessary and sufficient that the conditions

[
h_\lambda (h_\lambda-1){h_\mu\gamma^{i_\lambda}{k\mu l_\lambda}-(h_\mu-1)\delta^{i_\lambda}{k\mu l_\lambda}}=0
\tag{4}
]

be fulfilled, and that the quantities (\Gamma) be determined from the conditions

[
\Gamma^a_{db}=\gamma^a_{db},\qquad
\Gamma^a_{i_\lambda b}=h_\lambda\gamma^a_{i_\lambda b}+(1-h_\lambda)C^a_{i_\lambda b};
]

[
\Gamma^{i_\lambda}{k\mu l_\lambda}=
\begin{cases}
(h_\lambda-h_\mu)C^{i_\lambda}{k\mu l_\lambda}+h_\mu\gamma^{i_\lambda}{k\mu l_\lambda}, & \text{if } h_\lambda\ne 0,\
-h_\mu C^{i_\lambda}{k\mu l_\lambda}+(h_\mu-1)\delta^{i_\lambda}{k\mu l_\lambda}, & \text{if } h_\lambda=0.
\end{cases}
\tag{5}
]

We shall regard the space of the group (G) as the base space of the fibration of the group (G_\ell\times g_{(1)\mathrm{pr}}) by left cosets with respect to the subgroup (\bar g_{(1)}).

Every parametrized curve of the group (G) may be regarded as the projection of some curve of the space (G_\ell\times g_{(1)\mathrm{pr}}) satisfying system a). Indeed, choosing, in a neighborhood of an arbitrary point, the parameters (u^a,u^{i_\lambda},v^{i_\lambda}) so that (u^a,u^{i_\lambda}) are integrals of the system (\omega^a=0,\ \bar\omega^{i_\lambda}=0), prescribing them by the equations (u^a=u^a(t),\ u^{i_\lambda}=u^{i_\lambda}(t)), and substituting them into a), we obtain, for determining (v^{i_\lambda}), a system of ordinary differential equations which, evidently, has solutions. It is proved analogously that every curve of the space (G) is the projection of a curve of the space (G_\ell\times g_{(1)\mathrm{pr}}) along which (\theta^{i_\lambda}=0). Comparing equations c) and (1), we note that c) are the equations of the geodesic lines of the affine connection of the space (G) (where (G_\ell\times g_{(1)\mathrm{pr}}) is taken as the group of motions, and (\theta^{i_\lambda}=0) as the adjoining subspace). Hence every line having the same projection into the space (G) as some solution of system c) is itself a solution of this system. In particular, the general solutions of systems a) and c) have the same projections onto (G) as do the lines along which

[
d_t\omega^a=\Gamma^a_{i_\lambda b}\omega^{i_\lambda}\omega^b+\frac12\Gamma^a_{db}\omega^d\omega^b,
]

[
d_t\omega^{i_\lambda}=\Gamma^{i_\lambda}{k\mu l_\lambda}\omega^{k_\mu}\omega^{l_\lambda},
\tag{6}
]

and (\Gamma) are determined from (5). In other words, (6) are the equations of the projections of the general solutions of a), b) onto (G) in the basis of the forms (\omega^a,\omega^{i_\lambda}).

Let us now observe that equations (6), as in general all equations of the form ((b_1),(b_2)), preserve their structure if, instead of the group (g_{(1)}), one considers the group (g_{(2)}), i.e., assigns the forms (\omega^{i_1}) to the class of the forms (\omega^a). We shall now consider the space (G_\ell\times g_{(2)\mathrm{pr}}) and in it a system of the form a) with new parameters (h_{1\lambda}) ((\lambda=2,\ldots,n)). We supplement equations (6) by equations of the form ((b_3)) with respect to the invariant forms of the group (g_{(2)\mathrm{pr}}) in such a way that the coefficients of the equations are related to (6) by conditions of the form (4). Proceeding with the system thus obtained as above with equations a), b), we arrive at equations of the same form as equations (6), but depending on a larger number of parameters. The same constructions can be carried further with the aid of the subgroups (g_{(3)},\ldots,g_{(n)}), obtaining each time systems of equations depending on an ever larger number of parameters.

1. Let the initial system ((b_1),(b_2)) have the form (3). We shall obtain from this system new ones, as indicated in § 3, taking each time, as additional equations ((b_3)), equations of the same structure (3) in the group (g_{(\lambda)}) with subgroups (g_{(\mu)},\mu>\lambda). Evidently, this can always be done

can be done without contradicting the conditions of the form (4). Then all the newly obtained equations, for any choice of the parameters entering them, will be equations of the form (3). In particular, one may start from the equations of the geodesic connections of Cartan of the group (G), i.e. initially set the right-hand sides of the equations ((b_1)), ((b_2)) equal to zero. Using systems of the form a) with (h_\lambda \ne 0), (h_{1\lambda} \ne 0,\ldots), we obtain a family of equations of the form (3), depending on (n(n+1)/2) parameters, i.e. on the same number as the entire family (3) with arbitrary (\xi_\lambda,\xi_{\mu\lambda}). A more exact count shows that, allowing also zero values for (h_\lambda,h_{1\lambda},\ldots), one can in this way obtain any system of the form (3).

Thus we obtain a geometric characterization of the solutions of the system (3), i.e. of the geodesic lines of the corresponding connections. Indeed, in § 1 the geometric characterization of these lines was indicated for (n=1). Let us apply induction on the number (n) of subgroups and assume that the solutions of systems of the form (3) for the number of subgroups (\leq n-1) have been studied. Then, as follows from the preceding, the solutions of such a system for the case of (n) subgroups are obtained by repeated application of: 1) considering, in the direct products (G_\lambda \times g_{(\lambda)\mathrm{pr}}), lines whose projections onto the direct factors (G_\lambda) and (g_{(\lambda)\mathrm{pr}}) have already been studied; 2) selecting from all such lines a completely determined subfamily; and 3) projecting the selected lines into the space (G) as into the base space of the fibration of the group (G_\lambda \times g_{(\lambda)\mathrm{pr}}) by the cosets with respect to the subgroup (\bar g_{(\lambda)}).

1. Let the subgroup (g_0) of the group (G) be contained in (g_{(n)}) and be equipped with the subspace (H_0 \supset H_n). Then every connection with geodesics (3) determines a torsion-free connection in the space (G/g_0), whose geodesics are in the same relation to the geodesics (3) as for the connections (\Gamma'\xi) and (\Gamma\xi) in § 1.

Moscow State University
named after M. V. Lomonosov

Received
20 XI 1959

REFERENCES

1. A. M. Vasil’ev, Izv. Vyssh. Uchebn. Zaved., Mathematics, No. 2(9), 41 (1959).
2. P. K. Rashevskii, Tr. Seminar on Vector and Tensor Analysis, 9, 49 (1952).