Some Riemannian manifolds are expressed as a product manifold. Recently, I have read two articles about space-times. In both articles, the authors prove that a Riemannian manifold $\bar{M}^n$ is expressed as a product of the form $\mathbb R\times M^{n-1}$ i.e. the manifold splits off a factor $\mathbb{R}$. Both authors use similar techniques, namely integrable distribution, in this decomposition. Really, I do not understand this technique. Also, there are many splitting theorems in the literature of the same form, I mean $\mathbb R\times M^{n-1}$. 


For me, it is enough to know a characterization of Riemannian manifolds $M^{n+m}$ which we can express them as a product manifold $M\_1^n\times M\_2^m$. 


**Q1** Does this characterization exist? 


**Q2** What conditions could one use to characterize these manifolds? 


Note:


I asked the same question 2 days ago on Mathematics Stack exchange, I have got neither answer nor comment.



Is there a characterization of Riemannian manifolds that split off two factors?
Ralf Ponge and Helmut Reckziegel have show in their paper: "Twisted products in pseudo-Riemannian geometry" Geometriae Dedicata, October 1993, Volume 48, Issue 1, pp 15-25, the following result


Let $(M,g)$ be a simply connected psudo-Riemannian manifold with two complementary 
foliations $L$ and $K,$ whose leaves intersect perpendiculary. If the leaves of
$L$ are totally geodesic and geodesically complete and the leaves of $K$ are totally geodesic, then $(M,g)$ is isometric to a Riemmannian product, $(M\_1\times M\_2,g\_1\oplus g\_2),$ sucht that $L$ and $K$ correspond to the canonical foliations of the product.


Even more, they gave conditions on the foliations $L$ and $K$ in order to $(M,g)$ to be isometric to a twisted product (leaves of $L$ totally geodesic and leaves of $K$ totally umbilical) or a warped product (leaves of $L$ totally geodesic and leaves of $K$ extrinsic spheres). 


