MATHEMATICS

Yu. I. LEVIN

ON SOME SPACES WITH AFFINE CONNECTION ADMITTING MOTIONS

(Presented by Academician P. S. Aleksandrov on 29 XI 1960)

In this note we consider spaces with affine connection \(A_n\) that admit simply transitive, as well as Abelian, groups of affine collineations.

1. Theorem 1. The most general torsion-free \(A_n\) admitting a simply transitive group of affine collineations with operators \(X_\alpha = \underset{(\alpha)}{\xi^i}\dfrac{\partial}{\partial x^i}\) \((i,j,k,\alpha,\beta,\gamma = 1,2,\ldots,n)\) has the form

\[ \Gamma^k_{ij} = \frac{1}{2}\,\underset{(\alpha)}{\lambda^k} \left( \frac{\partial \underset{(\alpha)}{\lambda_i}}{\partial x^j} + \frac{\partial \underset{(\alpha)}{\lambda_j}}{\partial x^i} \right) + k^\alpha_{\beta\gamma}\, \underset{(\alpha)}{\lambda^k}\, \underset{(\beta)}{\lambda_i}\, \underset{(\gamma)}{\lambda_j}, \tag{1} \]

where \(\underset{(\alpha)}{\lambda^i}\) are the coefficients of the operators of the group reciprocal to the group \(X_\alpha\);

\[ \underset{(\alpha)}{\lambda_i}\,\underset{(\alpha)}{\lambda^j} = \delta^j_i, \qquad k^\alpha_{\beta\gamma}=k^\alpha_{\gamma\beta} \]

are arbitrary constants.

Proof. If \(A_n\) admits a group of collineations with operators \(X_\alpha\), then the connection object satisfies the system of equations \(D_L\Gamma^k_{ij}=0\), where \(D_L\) is the Lie derivative with respect to the vectors \(\underset{(\alpha)}{\xi^i}\), or, equivalently, the system

\[ \frac{\partial^2 \underset{(\alpha)}{\xi^k}}{\partial x^i \partial x^j} + \frac{\partial \Gamma^k_{ji}}{\partial x^l}\,\underset{(\alpha)}{\xi^l} + \Gamma^k_{lj}\, \frac{\partial \underset{(\alpha)}{\xi^l}}{\partial x^i} + \Gamma^k_{il}\, \frac{\partial \underset{(\alpha)}{\xi^l}}{\partial x^j} - \Gamma^l_{ij}\, \frac{\partial \underset{(\alpha)}{\xi^k}}{\partial x^l} =0. \tag{2} \]

Taking into account that

\[ \underset{(\alpha)}{\xi^l}\, \frac{\partial \underset{(\beta)}{\lambda^k}}{\partial x^l} - \underset{(\beta)}{\lambda^l}\, \frac{\partial \underset{(\alpha)}{\xi^k}}{\partial x^l} =0, \]

and passing in (2) to the vectors \(\underset{(\alpha)}{\lambda^i}\) of the reciprocal group, we obtain for \(\Gamma^k_{ij}\) the system

\[ \frac{\partial \Gamma^k_{ij}}{\partial x^l} = \Gamma^k_{im}\,\underset{(\alpha)}{\lambda^m} \frac{\partial \underset{(\alpha)}{\lambda_j}}{\partial x^l} + \Gamma^k_{mj}\,\underset{(\alpha)}{\lambda^m} \frac{\partial \underset{(\alpha)}{\lambda_i}}{\partial x^l} - \Gamma^m_{ij}\,\underset{(\alpha)}{\lambda^k} \frac{\partial \underset{(\alpha)}{\lambda_m}}{\partial x^l} - \]

\[ - \underset{(\alpha)}{\lambda^m}\, \underset{(\beta)}{\lambda^k}\, \frac{\partial \underset{(\alpha)}{\lambda_j}}{\partial x^l}\, \frac{\partial \underset{(\beta)}{\lambda_i}}{\partial x^m} - \underset{(\alpha)}{\lambda^m}\, \underset{(\beta)}{\lambda^k}\, \frac{\partial \underset{(\alpha)}{\lambda_i}}{\partial x^l}\, \frac{\partial \underset{(\beta)}{\lambda_m}}{\partial x^j} + \underset{(\alpha)}{\lambda^k}\, \frac{\partial^2 \underset{(\alpha)}{\lambda_i}}{\partial x^j \partial x^l}. \tag{3} \]

The symmetry conditions in \(i,j\) and the integrability conditions of system (3) are satisfied identically by virtue of the Maurer—Cartan equations. The object

\(\Gamma^k_{ij}\) therefore exists with arbitrary \(n^2(n+1)/2\) constants. By direct verification we are convinced that an object of the form (1) satisfies the system (3); but this object contains \(n^2(n+1)/2\) arbitrary constants, i.e. is the general solution.

If in the connection (1) \(k^\alpha_{\beta\gamma}=0\), then we obtain a space of affine connection without torsion, defined (in Cartan’s sense (1)) simply by a transitive group—the so-called group space.

Let us note that the connection (1) is equiaffine if and only if \(c^\alpha_{\beta\gamma} k_\alpha=0\), where \(c^\alpha_{\beta\gamma}\) are the structural constants of the group, and \(k_\alpha=k^\beta_{\alpha\beta}\).

From the theorem proved it easily follows the well-known fact that all \(A_n\) without torsion, admitting a simply transitive Abelian group of affine collineations, and only such spaces, have, in some coordinate system, constant components of the connection object. Indeed, if the group of collineations is Abelian, then the reciprocal group is also Abelian, whence it follows that

\[ \frac{\partial \overset{(\alpha)}{\lambda}_{[i}}{\partial x^{j]}}=0. \]

Therefore (1) can be represented in the form

\[ \frac{\partial \overset{(\alpha)}{\lambda}}{\partial x^k}\,\Gamma^k_{ij}(x) = \frac{\partial^2 \overset{(\alpha)}{\lambda}}{\partial x^i \partial x^j} + k^\alpha_{\beta\gamma} \frac{\partial \overset{(\beta)}{\lambda}}{\partial x^i} \frac{\partial \overset{(\gamma)}{\lambda}}{\partial x^j}, \qquad \text{where } \overset{(\alpha)}{\lambda}_i= \frac{\partial \overset{(\alpha)}{\lambda}}{\partial x^i}. \]

Putting \(y^\alpha=\overset{(\alpha)}{\lambda}(x)\), we obtain that in the system \((y^\alpha)\)
\(\Gamma^\alpha_{\beta\gamma}(y)=k^\alpha_{\beta\gamma}\).

1. The connection (1) can be represented in the form
   \[ \Gamma^k_{ij}=\overline{\Gamma}^k_{ij} + k^\alpha_{\beta\gamma}\lambda^k_{\alpha} \overset{(\beta)}{\lambda}_i \overset{(\gamma)}{\lambda}_j, \]
   where \(\overline{\Gamma}^k_{ij}\) is the group connection defined by the group of affine collineations of the connection \(\Gamma^k_{ij}\) (or, what is the same, by the reciprocal group). Let us note that the trajectories of the affine collineations of the space (1) are geodesics of the connection \(\overline{\Gamma}^k_{ij}\). Indeed, from the equations of the trajectories
   \[ \frac{dx^i}{dt}=c^\alpha \overset{(\alpha)}{\xi}{}^i \]
   we obtain
   \[ \frac{d^2x^k}{dt^2} = -\frac{1}{2}\lambda^k_{\alpha} \left( \frac{\partial \overset{(\alpha)}{\lambda}_{(i}}{\partial x^{j)}} \right) \frac{dx^i}{dt}\frac{dx^j}{dt} = -\overline{\Gamma}^k_{ij} \frac{dx^i}{dt}\frac{dx^j}{dt}. \]

Theorem 2. The affine collineations of the space (1) are translations if and only if
\[ k^\alpha_{\beta\gamma}=\delta^\alpha_{(\beta}e_{\gamma)}. \tag{4} \]

Indeed, from the remark made above it follows that the affine collineations of the connection \(\Gamma^k_{ij}\) are translations if and only if the connections \(\Gamma^k_{ij}\) and \(\overline{\Gamma}^k_{ij}\) have common geodesics, i.e. are related by the relations
\[ \Gamma^k_{ij}-\overline{\Gamma}^k_{ij} = \delta^k_{(i}\tau_{j)}, \]
where \(\tau_j\) is some vector. Hence, i.e. from the system
\[ k^\alpha_{\beta\gamma}\lambda^k_{\alpha} \overset{(\beta)}{\lambda}_i \overset{(\gamma)}{\lambda}_j = \delta^k_{(i}\tau_{j)}, \]
we easily obtain the required result.

Thus, the connection (1), under the condition (4), is the most general connection admitting a simply transitive group of translations.

1. Let a tensor field \(T\) be given on a manifold. The group of transformations of the manifold into itself leaving this tensor field invariant will be called the group of automorphisms of the tensor \(T\).

Theorem 3. Any tensor admitting a simply transitive group of automorphisms with operators
\[ X_\alpha=\overset{(\alpha)}{\xi}{}^i\frac{\partial}{\partial x^i} \]
has the form
\[ T^{i_1\ldots i_p}_{j_1\ldots j_q} = c^{\alpha_1\ldots \alpha_p}_{\beta_1\ldots \beta_q} \lambda^{i_1}_{\alpha_1}\cdots \lambda^{i_p}_{\alpha_p} \overset{(\beta_1)}{\lambda}_{j_1}\cdots \overset{(\beta_q)}{\lambda}_{j_q}, \tag{5} \]

where \(\lambda_{(\alpha)}^{i}\) are the coefficients of the operators of the reciprocal group, \(\lambda_{(\alpha)}^{i}\lambda_{j}^{(\alpha)}=\delta_{j}^{i}\), and \(c_{\beta_1\ldots\beta_q}^{\alpha_1\ldots\alpha_p}\) are arbitrary constants.

Indeed, the automorphism group of the tensor \(T\) is determined by the system
\(\underset{L}{D}T=0\), or

\[ \frac{\partial T^{i_1\ldots i_p}_{j_1\ldots j_q}}{\partial x^k}\xi^k +T^{i_1\ldots i_p}_{k j_2\ldots j_q}\frac{\partial \xi^k}{\partial x^{j_1}} +\cdots+ T^{i_1\ldots i_p}_{j_1\ldots j_{q-1}k}\frac{\partial \xi^k}{\partial x^{j_q}} - T^{k i_2\ldots i_p}_{j_1\ldots j_q}\frac{\partial \xi^{i_1}}{\partial x^k} -\cdots- T^{i_1\ldots i_{p-1}k}_{j_1\ldots j_q}\frac{\partial \xi^{i_p}}{\partial x^k} =0. \tag{6} \]

If the tensor admits a simply transitive group of automorphisms, then from (6), passing to the vectors of the reciprocal group, we obtain

\[ \frac{\partial T^{i_1\ldots i_p}_{j_1\ldots j_q}}{\partial x^k} = T^{i_1\ldots i_p}_{m j_2\ldots j_q}\lambda_{(\alpha)}^{m} \frac{\partial \lambda_{j_1}^{(\alpha)}}{\partial x^k} +\cdots+ T^{i_1\ldots i_p}_{j_1\ldots j_{q-1}m}\lambda_{(\alpha)}^{m} \frac{\partial \lambda_{j_q}^{(\alpha)}}{\partial x^k} - \]

\[ - T^{m i_2\ldots i_p}_{j_1\ldots j_q}\lambda_{(\alpha)}^{i_1} \frac{\partial \lambda_{m}^{(\alpha)}}{\partial x^k} -\cdots- T^{i_1\ldots i_{p-1}m}_{j_1\ldots j_q}\lambda_{(\alpha)}^{i_p} \frac{\partial \lambda_{m}^{(\alpha)}}{\partial x^k}. \]

This system, as may be verified, is completely integrable, and its general solution depends on \(n^{p+q}\) constants; but the tensor of the form (5), depending precisely on \(n^{p+q}\) constants, is a solution.

From the fact that for an Abelian simply transitive group in some coordinate system \(\xi_{(\alpha)}^{i}=\delta_{\alpha}^{i}\), we obtain:

Corollary. Tensors admitting a simply transitive Abelian group of automorphisms, and only such tensors, have constant components in some coordinate system.

It follows directly from Theorem 3 that the fundamental tensor of a Riemannian space \(V_n\) admitting a simply transitive group of motions has the form \(g_{ij}=c_{\alpha\beta}\lambda_i^{(\alpha)}\lambda_j^{(\beta)}\), where \(c_{\alpha\beta}=c_{\beta\alpha}\) are arbitrary constants. As is easy to verify, the connection of such a space has the form (1), where

\[ k_{\beta\gamma}^{\alpha} = c^{\alpha\delta}\bigl(c_{\varepsilon\beta}c_{\delta\gamma}^{\varepsilon} + c_{\varepsilon\gamma}c_{\delta\beta}^{\varepsilon}\bigr) \tag{7} \]

(\(c_{\beta j}^{\alpha}\) are the structural constants of the group of motions), and conversely, if for the connection (1) \(k_{\beta\gamma}^{\alpha}\) has the form (7), then this connection is Riemannian.

Theorem 3 can also be applied to spaces of almost symplectic connection \((2,3)\): the fundamental tensor of a space of almost symplectic connection admitting a simply transitive group of motions has the form \(a_{ij}=c_{\alpha\beta}\lambda_i^{(\alpha)}\lambda_j^{(\beta)}\) \((c_{(\alpha\beta)}=0)\), and the connection has the form (1), where \(k_{\beta\gamma}^{\alpha}\) satisfy the system
\(k_{\gamma[\beta}^{\delta}c_{\alpha]\delta} = \frac16\bigl(c_{\delta[\gamma}^{\delta}c_{\alpha]\gamma} + 2c_{\gamma\delta}^{\delta}c_{\alpha\beta}^{\delta}\bigr)\).
In particular, when the group is Abelian, we obtain a space with constant \(a_{ij}\) and \(\gamma_{ijk}\).

1. Theorem 4. Every tensor \(T^{i_1\ldots i_p}_{j_1\ldots j_s}\) admitting an Abelian group \(G_r\) of automorphisms with operators
   \(X_\alpha=\xi_{(\alpha)}^{i}\dfrac{\partial}{\partial x^i}\)
   \((i=1,2,\ldots,n;\ \alpha=1,2,\ldots,r)\), for which the rank
   \(\|\xi_{(\alpha)}^{i}\|=q\le r\), has, in some coordinate system, the form

\[ T^{i_1\ldots i_p}_{j_1\ldots j_s} = T^{i_1\ldots i_p}_{j_1\ldots j_s}(x^{q+1},\ldots,x^n). \tag{8} \]

Conversely, a tensor of the form (8) admits at least a \(q\)-parameter abelian group of automorphisms.

This is easily obtained from consideration of system (6) and from the fact that the coefficients of the operators of an abelian group can be reduced to the form \({}^{(4)}\)
\(\xi_{(a)}^{i}=\delta_a^i,\ \xi_{(m)}^{\alpha}=\psi_m^\alpha(x^{q+1},\ldots,x^n),\ \xi_{(m)}^\delta=0\)
\((\alpha=1,2,\ldots,q;\ \delta=q+1,\ldots,n;\ m=q+1,\ldots,r)\).

Hence, taking into account that no \(V_n\) admits an abelian group of motions with \(q<r\) \({}^{(4)}\), we obtain:

Corollary. A Riemannian space \(V_n\) admits an abelian group \(G_r\) \((q=r)\) of motions if and only if, in some coordinate system,

\[ g_{ij}=g_{ij}(x^{r+1},\ldots,x^n). \]

In particular, only a flat \(V_n\) admits a simply transitive abelian group of motions.

Consideration of system (2) shows that Theorem 4 is also valid for automorphisms of an object of affine connection, namely, the following holds:

Theorem 5. An affine connection without torsion, admitting an abelian group \(G_r\) of affine collineations
\(X_a=\xi_{(a)}^i \dfrac{\partial}{\partial x^i}\) with \(q\le r\), has, in some coordinate system, components of the form

\[ \Gamma_{ij}^k=\Gamma_{ij}^k(x^{q+1},\ldots,x^n). \tag{9} \]

Conversely, a connection of the form (9) admits at least a \(q\)-parameter abelian group of affine collineations.

1. Affine collineations of a space with a nonsymmetric affine connection \(G_{ij}^k\) are determined, as is known, by the system
   \(D_l\Gamma_{ij}^k=D_L\Omega_{ij}^k=0\), where
   \(\Gamma_{ij}^k=\dfrac12 G_{(ij)}^k,\ \Omega_{ij}^k\) is the torsion tensor; therefore, for a space admitting a simply transitive group of affine collineations, we obtain:
   \(\Gamma_{ij}^k\) has the form (1),
   \(\Omega_{ij}^k=l_{\beta\gamma}^{\alpha}\lambda_{(i)}^\beta\lambda_{j)}^\gamma\) (by Theorem 3), where
   \(l_{(\beta\gamma)}^\alpha=0\); thus \(G_{ij}^k\) has the form (1), where the symmetry requirements are no longer imposed on \(k_{\beta\gamma}^{\alpha}\). Theorem 2 also remains valid (only the relations (4) are replaced by \(k_{(\beta\gamma)}^\alpha=\delta_{(\beta}^\alpha e_{\gamma)}\)), as does the remark on a connection with constant components.

For an abelian group of collineations of a space with torsion we obtain, by Theorem 4,
\(\Omega_{ij}^k=\Omega_{ij}^k(x^{q+1},\ldots,x^n)\); therefore Theorem 5 is also valid for a connection with torsion.

Orekhovo-Zuevo Pedagogical Institute

Received
25 XI 1960

CITED LITERATURE

1. É. Cartan, Geometry of Lie Groups and Symmetric Spaces, Moscow, 1949.
2. V. G. Lemlein, DAN, 115, 655 (1957).
3. Yu. I. Levin, Scientific Reports of Higher Schools, Mathematics, 1, 42 (1959).
4. L. P. Eisenhart, Continuous Transformation Groups, Moscow, 1947.