  Let X be a compact Kaehler manifold. We expect that any direct sum
decomposition of the tangent bundle T(X) comes from a splitting of the
universal covering space of X as a product of manifolds, in such a way that the
given decomposition of T(X) lifts to the canonical decomposition of the tangent
bundle of a product. We prove this assertion when X is a Kaehler-Einstein
manifold or a Kaehler surface. Simple examples show that the Kaehler hypothesis
is necessary.
