We show how flatness and symmetry of an affinely connected manifold relate themselves to the symmetry of the curvature tensor and its covariant derivative in certain slots. Then we reduce Bianchi first identity to the same form as when the connexion is symmetric; even when the connexion is non-symmetric by considering other conditions.

With the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection.

In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.

The main result of this paper is to prove that a generalization of the Principle of Dependent Choices, introduced by A. Levy [2; see also 1, Chapter 8], is equivalent to a form of Zorn's Lemma.

A Sturmian comparison theorem is established for a pair of linear hyperbolic differential equations. While the equations may be time dependent (in the sense of not allowing a separation of variables), a measure of the strength of such time dependence enters into the hypotheses of the theorem.