Minimal coverings of manifolds with balls

For a closed manifold M, denote by C(M) the minimal number of balls which suffice to cover M. It is shown that C(M) coincides with the Ljusternik-Schnirelmann category cat M if the latter is not too low compared with the dimension of M. In this case it follows in particular that C(M) is an invariant of the homotopy type of M. One of the applications of this result is the following: Let M be a closed manifold of sufficiently high category. Then cat(M×S¹)=cat M+1. This is a partial affirmative answer to a long-standing conjecture.