MATHEMATICS

Yu. I. LEVIN

ON SPACES OF AFFINE CONNECTION ASSOCIATED WITH \(n\) VECTOR FIELDS

(Presented by Academician P. S. Alexandrov on 12 III 1960)

1. On an \(n\)-dimensional differentiable manifold let us consider \(n\) vector fields \(\overset{(\alpha)}{a_i}\) \((\alpha,\ i=1,2,\ldots,n)\), \(\left|\overset{(\alpha)}{a_i}\right|\ne 0\), and let
\(\overset{(\alpha)}{a_i}\overset{(\beta)}{a^i}=\delta^\alpha_\beta\),
\(\overset{(\alpha)}{a_i}\overset{(\alpha)}{a^j}=\delta_i^j\). We introduce a torsion-free affine connection satisfying the conditions
\(\nabla_k \overset{(\alpha)}{a_i}=\frac12\,\partial a_{[i}^{(\alpha)}/\partial x^{k]}\).
Obviously, such a connection is defined uniquely, and

\[ \Gamma_{ij}^{k}=\frac12\,\overset{(\alpha)}{a^k} \left( \frac{\partial \overset{(\alpha)}{a_i}}{\partial x^j} + \frac{\partial \overset{(\alpha)}{a_j}}{\partial x^i} \right). \]

We shall call it a \(B\)-connection, and denote the differentiable manifold on which it is defined by \(B_n\).

As is known, the object of a general torsion-free affine connection possessing \(n\) covariantly constant vector fields \(\overset{(\alpha)}{a_i}\) has the form

\[ G_{ij}^{k}=\overset{(\alpha)}{a^k}\, \frac{\partial \overset{(\alpha)}{a_j}}{\partial x^i}. \tag{1} \]

Thus, \(\Gamma_{ij}^{k}=\frac12 G_{(ij)}^{k}\), i.e. the \(B\)-connection is the associated torsion-free connection for a flat affine connection.

Theorem 1. An affine connection is a \(B\)-connection if and only if the equations of the geodesics have the form

\[ \frac{dx^i}{dt}=c^\alpha \overset{(\alpha)}{\xi^i}, \tag{2} \]

where \(t\) is an affine parameter; \(c^\alpha\) are arbitrary constants; \(\overset{(\alpha)}{\xi^i}\) are \(n\) vector fields.

Substituting (1) into the equations of geodesics
\[ \frac{d^2x^i}{dt^2}+\Gamma_{jk}^{i}\frac{dx^j}{dt}\frac{dx^k}{dt}=0, \]
we obtain
\(c^\alpha c^\beta \overset{(\beta)}{\xi^j}\nabla_j\overset{(\alpha)}{\xi^i}=0\) for arbitrary \(c^\alpha\), whence
\(\overset{(\beta)}{\xi^j}\nabla_j\overset{(\alpha)}{\xi^i} + \overset{(\alpha)}{\xi^j}\nabla_j\overset{(\beta)}{\xi^i}=0\);
after transformations we obtain

\[ \Gamma_{ij}^{k}=\frac12\,\overset{(\alpha)}{\xi^k}\, \frac{\partial \overset{(\alpha)}{\xi_{(i}}}{\partial x^{j)}} , \quad \text{where}\quad \overset{(\alpha)}{\xi_i}\overset{(\beta)}{\xi^i}=\delta^\alpha_\beta,\quad \overset{(\alpha)}{\xi_i}\overset{(\alpha)}{\xi^j}=\delta_i^j; \]

i.e. we have a \(B\)-connection. The necessity is easily verified.

Introduce the tensor
\[ S_{ij}^{k}=\frac12\,\overset{(\alpha)}{a^k}\, \frac{\partial \overset{(\alpha)}{a_{[i}}}{\partial x^{j]}} . \]
One can verify that the identity

\[ \nabla_{(i}S_{jk)}^{l}-S_{p(i}^{l}S_{jk)}^{p}=0 \tag{3} \]

holds. Taking into account that \(\Gamma_{ij}^{k}=G_{ij}^{k}-S_{ij}^{k}\), we easily obtain that
\(R_{ij,k}^{\ \ \ l}=\nabla_{[i}S_{j]k}^{l}-S_{m[i}^{l}S_{j]k}^{m}\), or, in view of (3),

\[ R_{ij,k}^{\ \ \ l}=S_{mk}^{l}S_{ij}^{m}-\nabla_k S_{ij}^{l}. \tag{4} \]

1. A space \(B_n\) with the property \(\nabla_k a=0\) \(\bigl(a=\det\|a_i^{(\alpha)}\|\bigr)\) will be called a space with invariant volume.

Let us introduce into consideration the vector \(S_i=S^k_{ik}\).

Theorem 2. The space \(B_n\) is equiaffine if and only if \(\nabla_{[i}S_{j]}=0\), and possesses invariant volume if and only if \(S_i=0\).

Indeed, \(R_{jk}=\nabla_k S_j-S^l_{mk}S^m_{jl}\), \(R_{[jk]}=\nabla_{[k}S_{j]}\), whence the first assertion follows. Further, we have

\[ \nabla_k a=\frac{1}{2}a\left(\frac{\partial\ln a}{\partial x^k} -a^l_{(\alpha)}\frac{\partial a_k^{(\alpha)}}{\partial x^l}\right); \]

but

\[ S_k=\frac{1}{2}\left(\frac{\partial\ln a}{\partial x^k} -a^i_{(\alpha)}\frac{\partial a_k^{(\alpha)}}{\partial x^i}\right) =\frac{\nabla_k a}{a}. \]

Thus every space with invariant volume is equiaffine.

We shall call the space \(\{a_i^{(\alpha)}\}\) conformal to the space \(\{\bar a_i^{(\alpha)}\}\), if

\[ \bar a_i^{(\alpha)}=e^\sigma a_i^{(\alpha)}; \]

if \(\sigma=\mathrm{const}\), then the spaces will be called similar. It is easy to see that a conformal mapping preserves equiaffineness.

Theorem 3. Among all mutually conformal equiaffine spaces \(B_n\) there is exactly one space with invariant volume (up to similarity).

Indeed,

\[ \bar S^k_{ij}=S^k_{ij}+\frac{1}{2}\delta^k_{(i}\sigma_{j)}, \]

therefore

\[ \bar S_i=S_i+\frac{1}{2}(n-1)\sigma_i \]

\((\sigma_i=\partial\sigma/\partial x^i)\), whence the assertion of the theorem follows.

1. The collection \(\{a_i^{(\alpha)}\}\) of vectors determining the space \(B_n\) will be called the fundamental coreper of the space; correspondingly \(\{a^i_{(\alpha)}\}\) will be called the fundamental reper. A fundamental coreper consisting of gradient vectors can be reduced to the form \(a_i^{(\alpha)}=\delta_i^\alpha\); the corresponding object \(\Gamma^k_{ij}=0\), and the space is flat. But not only a gradient coreper determines a flat space. Put

\[ \Gamma^k_{ij}\equiv \frac{1}{2}a^k_{(\alpha)} \left( \frac{\partial a_i^{(\alpha)}}{\partial x^j} + \frac{\partial a_j^{(\alpha)}}{\partial x^i} \right)=0; \]

then

\[ \frac{\partial a_i^{(\alpha)}}{\partial x^j} + \frac{\partial a_j^{(\alpha)}}{\partial x^i}=0, \]

whence

\[ \frac{\partial^2 a_i^{(\alpha)}}{\partial x^j\partial x^k} + \frac{\partial^2 a_j^{(\alpha)}}{\partial x^i\partial x^k}=0, \]

therefore

\[ \frac{\partial^2 a_j^{(\alpha)}}{\partial x^k\partial x^i} + \frac{\partial^2 a_k^{(\alpha)}}{\partial x^j\partial x^i}=0, \]

\[ \frac{\partial^2 a_k^{(\alpha)}}{\partial x^i\partial x^j} + \frac{\partial^2 a_i^{(\alpha)}}{\partial x^k\partial x^j}=0. \]

Subtracting the 2nd equality from the sum of the 1st and 3rd, we obtain

\[ \frac{\partial^2 a_i^{(\alpha)}}{\partial x^j\partial x^k}=0, \]

whence

\[ a_i^{(\alpha)}=c_{ij}^{\alpha}x^j+c_i^\alpha. \]

Theorem 4. Every flat \(B_n\) can be given in the form

\[ a_i^{(\alpha)}=c_{ij}^{\alpha}x^j+c_i^\alpha, \tag{5} \]

where \(c_i^\alpha,\ c_{ij}^\alpha=-c_{ji}^\alpha\) are constants.

1. In view of the fact that \(\Gamma^k_{ij}=\frac{1}{2}G^k_{(ij)}\), and that every flat connection can be reduced to the form \(G^k_{ij}=\delta^k_j\,\partial\ln\varphi_j/\partial x^i\) (1), we obtain that every \(B\)-connection can be reduced to the form

\[ \Gamma^k_{ij} = \frac{1}{2} \left( \delta^k_i\frac{\partial\ln\varphi_i}{\partial x^j} + \delta^k_j\frac{\partial\ln\varphi_j}{\partial x^i} \right). \]

The curvature tensor in this coordinate system has the form

\[ R_{lk,i}^{\ \ \ q} = \delta_i^q A_{lik} - \delta_k^q A_{kil}, \]

where

\[ A_{lik} = \frac{1}{4} \left( 2\frac{\partial^2\ln\varphi_l}{\partial x^i\partial x^k} + \frac{\partial\ln\varphi_l}{\partial x_k} \frac{\partial\ln\varphi_l}{\partial x_l} - \frac{\partial\ln\varphi_l}{\partial x^k} \frac{\partial\ln\varphi_k}{\partial x^i} - \frac{\partial\ln\varphi_l}{\partial x^i} \frac{\partial\ln\varphi_i}{\partial x^k} \right). \]

If the space is projectively flat and \(n>2\), then
\(R_{lk,i}^{\ q}=\delta_{[k}^{q}P_{l]i}+\delta_i^q P_{[lk]}\), where
\(P_{ki}=-(nR_{ki}+R_{ik})/(n^2-1)\). Let \(q=i\ne k,l\). Then \(P_{[lk]}=0\), i.e. \(R_{[lk]}=0\), and we obtain that for \(n>2\) every projectively flat \(B_n\) is equiaffine. For \(n=2\) this can also be proved. But every equiaffine projectively flat space can be given in the form
\(\Gamma_{ij}^{k}=\frac12\delta_{(i}^{k}\tau_{j)}\), where \(\tau_j=\partial\tau/\partial x^j\), i.e.
\(\partial \overset{(\alpha)}{a_i}/\partial x^j+\partial \overset{(\alpha)}{a_j}/\partial x^i=\overset{(\alpha)}{a_i}\tau_j+\overset{(\alpha)}{a_j}\tau_i\); introducing
\(\overset{(\alpha)}{A_i}=e^{-\tau}\overset{(\alpha)}{a_i}\), we obtain
\(\partial \overset{(\alpha)}{A}_{(i}/\partial x^{j)}=0\), whence
\(\overset{(\alpha)}{A_i}=c_{ij}^{\alpha}x^j+c_i^\alpha\), where \(c_{(ij)}^\alpha=0\). Thus:

Theorem 5. The fundamental coframe of any projectively flat \(B_n\) can be represented in the form
\[ \overset{(\alpha)}{a_i}=e^\tau\bigl(c_{ij}^\alpha x^j+c_i^\alpha\bigr), \tag{6} \]
where \(c_{ij}^\alpha=-c_{ji}^\alpha\), \(c_i^\alpha\) are constants; i.e. every projectively flat \(B_n\) is conformally flat, and conversely.

Corollary. Every equiprojective connection admits a fundamental coframe, i.e. is a \(B\)-connection.

Indeed, the equiprojective connection \(\Gamma_{ij}^{k}=\frac12\delta_{(i}^{k}\tau_{j)}\) admits the coframe (6) with the property
\(\nabla^k\overset{(\alpha)}{a_i}=\frac12\partial\overset{(\alpha)}{a}_{[i}/\partial x^{k]}\).

We shall call a space with a fundamental coframe of the form
\(\overset{(\alpha)}{a_i}=e^\tau\overset{(\alpha)}{A_i}\), where
\(\overset{(\alpha)}{A_i}=\partial\overset{(\alpha)}{A}/\partial x^i\), conformally gradient. Obviously, this is a special case of projectively flat \(B_n\). It is easy to see that for a conformally gradient \(B_n\)
\[ \frac{\partial \overset{(\alpha)}{a}_{[i}}{\partial x^{j]}} =\tau_{[j}\overset{(\alpha)}{a}_{i]}, \tag{7} \]
whence, contracting with \(a^j\), we obtain
\(\tau_j=\dfrac{2}{n-1}S_j\), and therefore
\[ S_{ij}^{k}=\frac{1}{n-1}S_{[i}\delta_{j]}^{k}. \tag{8} \]

Conversely, if conditions (8) are satisfied, and \(S_i\) is a gradient, then, putting
\(\tau_i=\dfrac{2}{n-1}S_i\), we obtain (7), whence, representing \(\overset{(\alpha)}{a_i}\) in the form \(e^\tau\overset{(\alpha)}{A_i}\), we obtain
\(\partial\overset{(\alpha)}{A}_{[i}/\partial x^{j]}=0\). Thus the following is true:

Theorem 6. The space \(B_n\) is conformally gradient if and only if \(S_i\) is a gradient and
\[ S_{ij}^{k}=\frac{1}{n-1}S_{[i}\delta_{j]}^{k}. \]

5. It is easy to see that spaces without torsion, defined by a simply transitive group (see (2); we shall call such spaces group spaces), form a subclass of the spaces \(B_n\); namely, the group spaces are those \(B_n\) for which the vectors \(\overset{(\alpha)}{a^i}\) of the fundamental frame are generators of a certain simply transitive group, i.e. satisfy the conditions
\[ \overset{(\alpha)}{a^k}\frac{\partial \overset{(\beta)}{a^i}}{\partial x^k} - \overset{(\beta)}{a^k}\frac{\partial \overset{(\alpha)}{a^i}}{\partial x^k} = c_{\alpha\beta}^{\gamma}\overset{(\gamma)}{a^i}, \tag{9} \]
where \(c_{\alpha\beta}^{\gamma}\) are constants,
\(c_{(\alpha\beta)}^\gamma=c_{\varepsilon(\alpha}^{\delta}c_{\beta\gamma)}^{\varepsilon}=0\).

Transforming (9), we obtain
\(2a_k\overset{(\beta)}{a^i}\overset{(\alpha)}{a^j}S_{ij}^{k}=c_{\alpha\beta}^{\gamma}\). Covariant differentiation of the scalars \(c_{\alpha\beta}^{\gamma}\) gives
\[ \nabla_k S_{ij}^{l}-S_{m(k}^{l}S_{ij)}^{m}=0, \tag{10} \]

whence, cycling, we obtain \(\nabla_{(k}S^l_{ij)}-3S^l_{m(k}S^m_{ij)}=0\); comparing with (3), we have

\[ S^l_{m(k}S^m_{ij)}=0, \tag{11} \]

whence, by virtue of (10),

\[ \nabla_k S^l_{ij}=0. \tag{12} \]

The conditions \(c^\gamma_{(\alpha\beta)}=0\) are satisfied, and the Jacobi identities lead to (11). Let us note that, owing to the identities (3), conditions (11) follow from (12). Conversely, if (11) holds, then, as is easy to see, the \(c^\gamma_{\alpha\beta}\) determined by the system (9) are constant, skew-symmetric, and satisfy the Jacobi identities. Thus, we have proved:

Theorem 7. In order that the space \(B_n\) be a group space, it is necessary and sufficient that conditions (12) hold.

From Theorem 7 and (4) there immediately follows the known fact: all group spaces are symmetric. Let us note that all these spaces are equiaffine, since their Ricci tensor \(R_{ij}=S^l_{kj}S^k_{li}\) is symmetric.

6. By a motion we shall mean a transformation of the space \(B_n\) preserving the fundamental coframe, i.e., one for which \(D_L a_i^{(\alpha)}=0\) (\(D_L\) is the Lie derivative), or

\[ \frac{\partial \xi^k}{\partial x^i}a_k^{(\alpha)} + \frac{\partial a_i^{(\alpha)}}{\partial x^k}\xi^k=0, \tag{13} \]

or, what is the same,

\[ \nabla_i\xi^k=S^k_{ij}\xi^j. \tag{14} \]

The integrability conditions for this system are
\((\nabla_k S^l_{ij}-S^l_{m(k}S^m_{ij)})\xi^k=0\), whence:

Theorem 8. Among the spaces \(B_n\), the group spaces and only they admit an \(n\)-parameter group of motions.

Thus, group spaces are characterized by maximal mobility,

Theorem 9. All motions of spaces \(B_n\) are translations.

Indeed, the motion \(x^{i'}=x^i+\xi^i\delta t\) is a translation, i.e. its trajectories are geodesics, if and only if
\[ \xi^k(\xi^l\nabla_k\xi^i-\xi^i\nabla_k\xi^l)=0, \]
as follows from the equations of geodesics. But for \(B_n\) this condition, by virtue of (14), is fulfilled, since
\[ \xi^k\nabla_k\xi^i=S^i_{kl}\xi^k\xi^l=0. \]

Let us consider, in conclusion, motions of conformally gradient spaces
\(a_i^{(\alpha)}=\delta_i^\alpha e^\sigma\). From (13)
\[ \partial \xi^l/\partial x^i+\delta_i^l\sigma_k\xi^k=0, \]
whence, as is easy to see, \(\xi^l=cx^l+c^l\), where the constants \(c,\ c^l\) must satisfy the equation

\[ \sigma_k(cx^k+c^k)+c=0. \tag{15} \]

The number of solutions \((c,c^k)\) of this equation gives the order of the group of motions.

If we now consider the conformally gradient space
\(A_i^{(\alpha)}=\delta_i^\alpha e^{2\sigma}\), then for it
\(\Gamma^k_{ij}=\delta^k_{(i}\sigma_{j)}\), and for such a space the number of solutions of equation (15) means the number of parallel fields of contravariant vectors (see \((^3)\)). Thus, we have proved:

Theorem 10. The groups of motions of conformally gradient spaces are subgroups of the affine group. Their order is equal to the number of parallel fields of contravariant vectors admitted by the space of the same type, but with the conformality factor squared.

Velikie Luki Pedagogical Institute

Received
11 III 1960

REFERENCES

1. L. P. Eisenhart, Non-Riemannian Geometry, 1927.
2. Л. П. Эйзенхарт, Непрерывные группы преобразований, 1947, Ch. V.
3. Ю. И. Левин, Научн. докл. высш. школы, No. 1 (1959).