M. A. ULANOVSKII

ON CONDITIONS DETERMINING OBJECTS OF AFFINE CONNECTION OF A RIEMANNIAN SPACE

(Presented by Academician A. N. Kolmogorov on 25 X 1956)

As is known, a given affine connection can be regarded as associated with a Riemannian or pseudo-Riemannian metric if and only if there exists, in this connection, a parallel field of the metric tensor \(g_{ij}\). The question of the existence of such a field reduces to the study of the integrability conditions of the equations

\[ Dg_{ij}=0 \qquad (i,j=1,\ldots,n), \tag{1} \]

where \(Dg_{ij}\) is the absolute differential of the metric tensor.

Below these conditions are given for the case \(n=3\); the smoothness class of the manifold \(X_3\) and of the given affine connection is assumed sufficiently high.

Let \(R_{ij}, \nabla_k R_{ij}, \nabla^2_{kl} R_{ij}\) \((i,j,k,l=1,2,3)\) be the matrices formed from the components of the curvature tensor and its covariant derivatives of the first and second orders (for example, the matrix \(R_{ij}\) is composed of the components \(R^l_{ij,k}\) of the curvature tensor, where \(k,l\) are the numbers of the matrix element; \(i,j\) are the numbers of the matrix).

If the given connection is Riemannian, then in the basic manifold \(X_3\) there exists a closed set without interior points, decomposing \(X_3\) into open connected sets, in each of which one of the following conditions is satisfied:

1) \(R_{ij}(x)=0\) \((i,j=1,2,3)\) at each point \(x\).

2) Among the matrices \(R_{ij}\) there exists one \(A(x)\), different from zero at each point; all the remaining matrices among \(R_{ij}, \nabla_k R_{ij}\) \((i,j,k=1,2,3)\) differ from it only by scalar multipliers.

3) Among the matrices \(R_{ij}\) there exists one \(A(x)\), different from zero at each point; the other two matrices among \(R_{ij}\) differ from it only by scalar multipliers, but among the matrices \(\nabla_k R_{ij}\) there exists a matrix \(B(x)\), linearly independent of \(A(x)\). All matrices among \(\nabla_k R_{ij}, \Delta^2_{kl} R_{ij}\) are linear combinations of the matrices \(A(x), B(x)\), and \(C(x)=A(x)B(x)-B(x)A(x)\).

4) Among the matrices \(R_{ij}\) one can choose two linearly independent matrices \(A(x), B(x)\); all the matrices \(R_{ij}, \nabla_k R_{ij}\) are linear combinations of the matrices \(A, B\), and \(C=AB-BA\).

Finally, at each point of \(X_3\) the following condition is satisfied: a) the matrix \(A(x)\) mentioned in condition 2) is similar to a skew-symmetric one; the matrices \(A(x)\) and \(B(x)\) mentioned in conditions 3), 4) can be simultaneously reduced to skew-symmetric form.

Theorem 1. If in some domain \(V\subset X_3\) one of the conditions 1), 2), 3), 4) is satisfied together with condition a), then there exists a field of a positive-definite metric tensor \(g_{ij}\) satisfying equations (1) in this domain.

We note that in the case of a pseudo-Riemannian metric condition a) must be modified accordingly.

The proof of this theorem is based on the following assertion concerning the holonomy groups of composite manifolds with a linear connection. Let \(L_k(x)\) be the minimal Lie algebra containing the matrices formed from the components of the curvature tensor and its covariant derivatives up to order \(k\), inclusive, at the point \(x\). Let \(p(x)\) be the smallest number such that \(L_p(x)=L_{p+1}(x)\).

Theorem 2. There exists a set \(M\), containing no interior points, which decomposes the base manifold of a space with a linear connection into domains such that, for each of them, the Lie algebra determining the identity component of the holonomy group of this domain coincides with \(L_p(x)\) at an arbitrary point \(x\).

This theorem is a very simple refinement of a known result of Nijenhuis (1); for its proof it is enough to observe that the algebra \(L_p(x)\) is constant (for the given connection) in some domain if \(p(x)\) and the dimension of \(L_p(x)\) are constant in it.

Finally, we note that the last theorem makes it possible to establish the integrability condition for equations (1) also in the general case of arbitrary \(n\), but the algebraic form of these conditions is very complicated.

REFERENCES CITED

1. A. Nijenhuis, Proc. Nederl. Akad. v. Wetensch. Amsterd., A 56 (3) (1953); A 57 (1) (1953).