De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$
that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the levi-Chevita connection is isometric to the direct product of two Riemanninan manifolds $M'\times M''$.
Question 1. In the first place I would like to have a good reference for a clear "modern" and complete proof of this theorem, if it exists (more recent than Kobayshi-Nomizu pp. 187-193) (Note, that Besse 10.44 claimed that no simple proof exists yet).
Eddited.
Question 2. Secondly it seems to me that there should be some statement much more general than de Rham theorem. Namely, suppose we have a metric space $X$ that is locally decomposable as an isometric product of two in such a way that this decomposition is "coherent" in a appropriate sense, i.e. forms something like a presheaf. When will we be able to say that $X=Y\times Z$? (I am interested only in the cases when this will work, not when this will fail). As corollary of such a general statement one should be able to deduce de Rham theorem for example, for Finsler of polyhedral manifold, ect.
Best Answer
For the first question, I can provide some literature:de Rham's original proof, Wu Hongxi's Ph.D thesis On the de Rham decomposition theorem. I don't know which reference would be earliest one. http://dspace.mit.edu/handle/1721.1/11601