MATHEMATICS

A. M. VASIL′EV

\(C'\)-CONNECTIONS IN HOMOGENEOUS SPACES AND THEIR TOTALLY GEODESIC SUBMANIFOLDS

(Presented by Academician P. S. Aleksandrov on 27 IV 1961)

A subgroup \(g\) of a Lie group \(G\) is called framed \((({}^{1}),\) cf. \(({}^{2}))\) if the Lie algebra \(\dot G\) of the group \(G\) is decomposed into a direct sum of subspaces \(\dot g\) and \(H\), where \(\dot g\) is the subalgebra corresponding to the subgroup \(g\), and \(H\) is invariant with respect to the corresponding \(\dot g\) subgroup \(\dot g^{*}\) of the group \(G^{*}\) attached to \(G\).

Let the right-invariant basic differential forms of the group \(G\) be divided into \(q+2\) groups \(\omega^{i_\lambda}, \omega^i\) \((\lambda = 0, 1, \ldots, q)\) in such a way that the structure equations of the group take the form

\[ D\omega^{i_\lambda} = C^{i_\lambda}_{k_\lambda l}[\omega^{k_\lambda}\omega^l] + C^{i_\lambda}_{k_\lambda l_\nu}[\omega^{k_\lambda}\omega^{l_\nu}] + \tfrac12 C^{i_\lambda}_{k_\mu l_\mu}[\omega^{k_\mu}\omega^{l_\mu}], \]
\[ D\omega^i = \tfrac12 C^i_{kl}[\omega^k\omega^l] + \tfrac12 C^i_{k_\lambda l_\lambda}[\omega^{k_\lambda}\omega^{l_\lambda}], \tag{1} \]

where \(\nu>\lambda,\ \mu<\lambda,\ \lambda=0,1,\ldots,q\).

This is equivalent (see \(({}^{1})\), item 5) to specifying in \(G\) a series of subgroups \(g_\lambda,\ g\) \((\lambda=1,\ldots,q;\ g_1 \supset g_2 \supset \cdots \supset g_q \supset g)\), framed by the subspaces \(H_\lambda, H\), with \(H \supset H_q \supset \cdots \supset H_1\).

Suppose that \(g\) contains no nontrivial invariant subgroup of the group \(G\), and denote by \(G/g\) the homogeneous space for which \(g\) is the stabilizer subgroup of one of its points \(M\).

1. Let the index \(\hat\lambda\) run through a certain (arbitrary) subset of the set of values \(0,1,\ldots,q\), and let \(\bar\lambda\) be the complementary subset. The Pfaffian system of equations in the space \(G/g\)

\[ \omega^{i_{\hat\lambda}} = 0 \tag{2} \]

is invariant with respect to all transformations of the group \(G\). In particular, the system of equations (2), in which \(\hat\lambda=0,1,\ldots,p-1\) \((p\le q)\), is completely integrable and determines in \(G/g\) systems of imprimitivity with respect to the subgroup \(g_p\). For example, the maximal integral manifold of this system passing through the point \(M\) is the set of points obtained from \(M\) under all transformations of the subgroup \(g_p\).

1. For any choice of constants \(\xi_{\nu\lambda}\) \((\nu,\lambda=0,1,\ldots,q;\ \nu>\lambda)\), the system of differential equations

\[ d_t\omega^{i_\lambda} = C^{i_\lambda}_{k_\lambda l}\omega^{k_\lambda}\omega^l + \xi_{\nu\lambda}C^{i_\lambda}_{k_\lambda l_\nu}\omega^{k_\lambda}\omega^{l_\nu}, \tag{3} \]

where \(d_t\omega=-dt\cdot d(\omega/dt)\), defines in \(G/g\) the geodesic lines of a completely determined torsion-free affine connection invariant with respect to the group \(G\). The family of connections obtained in this way depends, generally speaking, on \(q(q+1)/2\) parameters (but not fewer than on \(q\)). We shall call it the family of \(C'\)-connections of the space \(G/g\), determined by the series of subgroups

\(\mathfrak{g}^\lambda,\ \mathfrak{g}\), equipped with subspaces \(H_\lambda,\ H\). In paper \((^1)\) a \(q\)-parameter subfamily of this family was studied, called the family of \(C\)-connections.

1. The geodesics of every \(C'\)-connection that satisfy system (2) at some point satisfy it identically. In other words, system (2) determines in \(G/g\) a geodesic field of directions for every \(C'\)-connection with geodesics (3). In particular, all systems of imprimitivity by subgroups \(g_\lambda,\ \lambda=1,\ldots,q\), are completely geodesic submanifolds with respect to the corresponding family of \(C'\)-connections.

Consider the subfamily of the family of \(C'\)-connections for which all \(\xi_{\nu\mu}\) with \(\nu\in\bar\lambda,\ \mu\in\bar\lambda\) (see item 2) are equal to zero. This subfamily is characterized by the fact that all geodesics of its connections satisfying system (2) are trajectories of one-parameter subgroups of the group \(G\).

Let the set of values of the index \(\bar\lambda\), in turn, be divided into two subsets \(\lambda',\lambda''\) according to the criterion \(\lambda''\geq \lambda_0,\ \lambda'<\lambda_0\), where \(\lambda_0\) is a fixed integer. The equalities \(\xi_{\lambda'\lambda''}=1\) single out a subfamily of \(C'\)-connections having the property that the field of directions determined by the system

\[ \omega^{i\hat\lambda}=0,\qquad \omega^{i\lambda'}=0 \tag{4} \]

is parallel along all paths satisfying the system of equations

\[ \omega^{i\hat\lambda}=0,\qquad \omega^{i\lambda''}=0. \tag{5} \]

In other words, any vector of the space \(G/g\) satisfying system (4) at the initial point, under parallel displacement along a path satisfying system (5), will satisfy system (4) at every point of the path.

In particular, the family of \(C'\)-connections singled out by the conditions \(\xi_{\nu\mu}=1\) for \(\nu\geq p,\ \mu<p\), is characterized by the absolute parallelism of the systems of imprimitivity by the subgroup \(g_p\).

The propositions formulated have a number of consequences. Take a geodesic line of a \(C'\)-connection satisfying system (2), for which \(\xi_{\lambda'\lambda''}=1\). Consider the natural projection of this line into the homogeneous space \(G/g_p\), where \(p\) is greater than all values \(\lambda'\) and not greater than the least of the values \(\lambda''\) (for example, \(p=\lambda_0\)). This projection will be a geodesic line of one of the \(C'\)-connections of the space \(G/g_p\), specified in it by the series of subgroups \(g_1,\ldots,g_p\), equipped with subspaces \(H_1,\ldots,H_p\).

In particular, the natural projection of every geodesic of every \(C'\)-connection for which \(\xi_{\nu\mu}=1\) for \(\nu\geq\lambda_0,\ \mu<\lambda_0\), into the space \(G/g_{\lambda_0}\), is a geodesic \(C'\)-connection of this space.

1. We shall call a subgroup \(\mathrm{Li}\,f\) of the group \(G\) normal to the subgroup \(g\) in its equipment \(H\) \((^3)\) if the corresponding subalgebra \(\mathrm{Li}\,\hat f\) decomposes into the direct sum of its intersections with \(H\) and with the subalgebra \(\hat g\). The following propositions hold.

Let the subgroup \(f\) be normal to all subgroups \(g_\lambda,\ g\) forming the series (see item 1) in their equipments \(H_\lambda,\ H\). Subjecting a point \(M\) of the space \(G/g\) to all transformations of the subgroup \(f\), we obtain in \(G/g\) a submanifold completely geodesic for all \(C'\)-connections determined by the series of equipped subgroups \(g_\lambda,\ g\).

Let, moreover, \(f\) be an invariant subgroup of the group \(G\). Denote by \(\bar g_\lambda,\bar g\) the images of the subgroups \(g_\lambda,\ g\) under the natural homomorphism of \(G\) onto \(G/f=\bar G\). Denote by \(\bar H_\lambda,\bar H\) the images of the subspaces \(H_\lambda,\ H\) under the corresponding homomorphism of algebras \(\hat G\to\hat{\bar G}\). In our case the subspaces \(\bar H_\lambda,\bar H\) determine equipments of the subgroups \(\bar g_\lambda,\bar g\) of the group \(\bar G\). Every geodesic \(C'\)-connection of the space \(G/g\), under the natural mapping of \(G/g\) onto \(\bar G/\bar g\), passes into a geodesic \(C'\)-connection of the latter

space, determined by a series of subgroups \(\bar g_\lambda, \bar g\), equipped with subspaces \(\bar H_\lambda, \bar H\).

1. Let, as in Sec. 1 with \(q=0\), the basic forms \(\omega^{i_0}, \omega^i\) of the group \(G\) be chosen in such a way that the equations \(\omega^{i_0}=0\) determine the subgroup \(g\), and the equations \(\omega^i=0\) its equipping space \(H\). Denote by \(\Phi(g,H)\) the subgroup of the group \(G\) generated by all elements belonging to one-parameter subgroups along which the equations \(\omega^i=0\) are satisfied. \(\Phi(g,H)\) is an invariant subgroup. The equations defining the corresponding Lie subalgebra can be obtained by setting equal to zero all linear combinations of the forms \(\omega^i\) whose exterior differentials do not contain the forms \(\omega^{i_0}\). We denote the intersection \(\Phi(g,H)\cap g\) by \(\varphi(g,H)\). The following propositions hold.

Let \(f_{\lambda_0}\) be a subgroup of the group \(G\), normal to the subgroup \(g_{\lambda_0}\) from the series \(g_\lambda, g\) in the equipment \(H_{\lambda_0}\), and let \(f\) be a subgroup of the group \(g_{\lambda_0}\), normal to the subgroups \(g_\mu, g\) \((\mu=\lambda_0+1,\ldots,q)\) in the equipments \(H_\mu\cap \hat g_{\lambda_0}, H\cap \hat g_{\lambda_0}\), and containing \(\varphi(g_{\lambda_0}\cap f_{\lambda_0}, H_{\lambda_0}\cap \hat f_{\lambda_0})\). Denote by \(P\) the set of points of the space \(G/g\) obtained from a point \(M\) by transformations of the group \(f\). Subjecting all points of the manifold \(P\) to transformations of the group \(\Phi(g_{\lambda_0}\cap f_{\lambda_0}, H_{\lambda_0}\cap \hat f_{\lambda_0})\), we obtain a submanifold of the space \(G/g\), totally geodesic with respect to all \(C'\)-connections for which the coefficients \(\zeta_{\nu\mu}=1\) for \(\nu\geq \lambda_0,\ \mu<\lambda_0\).

The natural generalization of this proposition is formulated as follows. Let \(\lambda_1,\lambda_2,\ldots,\lambda_s\) be a certain subset of the set of numbers \(0,1,\ldots,q\), arranged in increasing order. Let the subgroup \(f_{\lambda_1}\) of the group \(G\) be normal to \(g_{\lambda_1}\) in the equipment \(H_{\lambda_1}\); let the subgroup \(f_{\lambda_2}\) of the group \(g_{\lambda_1}\) contain \(\varphi(f_{\lambda_1}\cap g_{\lambda_1}, \hat f_{\lambda_1}\cap H_{\lambda_1})\) and be normal to the subgroups \(g_\mu\), \(\lambda_1<\mu\leq \lambda_2\); let the subgroup \(f_{\lambda_3}\) of the group \(g_{\lambda_2}\) contain \(\varphi(f_{\lambda_2}\cap g_{\lambda_2}, \hat f_{\lambda_2}\cap H_{\lambda_2})\) and be normal to the subgroups \(g_\mu\), \(\lambda_2<\mu\leq \lambda_3\), and so on; finally, let the subgroup \(f\) of the group \(g_{\lambda_s}\) contain \(\varphi(f_{\lambda_s}\cap g_{\lambda_s}, \hat f_{\lambda_s}\cap H_{\lambda_s})\) and be normal to the subgroups \(g_\mu, g\), \(\lambda_s<\mu\leq q\). Subjecting the point \(M\) to all transformations of the group \(f\), the resulting manifold to all transformations of the group \(\Phi(f_{\lambda_s}\cap g_{\lambda_s}, \hat f_{\lambda_s}\cap H_{\lambda_s})\), the newly obtained manifold to all transformations of the group

\[ \Phi(f_{\lambda_{s-1}}\cap g_{\lambda_{s-1}}, \hat f_{\lambda_{s-1}}\cap H_{\lambda_{s-1}}) \]

and so on up to and including the group \(\Phi(f_{\lambda_1}\cap g_{\lambda_1}, \hat f_{\lambda_1}\cap H_{\lambda_1})\), we obtain in the space \(G/g\) a submanifold totally geodesic with respect to all \(C'\)-connections for which all coefficients \(\zeta_{\nu\mu}\) satisfying the conditions \(\nu\geq \lambda_a>\mu\) for at least one of the numbers \(\lambda_1,\lambda_2,\ldots,\lambda_s\) are equal to one.

Moscow State University
named after M. V. Lomonosov

Received
21 IV 1961

References

1. A. M. Vasil’ev, Izv. Vyssh. uchebn. zaved., Mathematics, No. 2, 41 (1959).
2. P. K. Rashevsky, Tr. seminara po vektorn. i tenzorn. analizu, 9, 49 (1952).
3. A. M. Vasil’ev, DAN, 128, No. 2, 223 (1959).