Suppose that $S$ and $T$ are two smooth manifolds and '$ \Re$' be the reals with the normal manifold structure. And here I use '$=$' to mean diffeomorphism. Is the statement below true?

$ S \times \Re = T \times \Re \Rightarrow S = T$.

What if '$=$' meant homeomorphism?

What if $S$ and $T$ are compact?

PS: Bartnik's splitting theorem states that if a space-time has a COMPACT cauchy hypersurface and is geodesically complete and has non negative ricci curvature every where then it can be split isometrically as a 3-manifold with a riemannian metric and $ \Re$ with the metric $-d^2t$.

It has been shown that such space-times can be splitt smoothly, so if there is a unique splitting then one can reduce the problem to something like:

if such and such conditions hold can the section $s:S \times \Re \longrightarrow E(S) \oplus E(\Re)$ be written as $s1 \oplus s2$ where $s1: S \longrightarrow E(S)$ and $s2: \Re \longrightarrow E(\Re)$.

i know that it really won't get transfered to this exactly. for example $f(x,y)dx + g(x,y)dy$ can be written in the form $F(U)dU + g(V)dV$ iff $ \frac{\partial{f}}{\partial y } = \frac{\partial{g}}{\partial x }$.

(i know that this is a very crude reasoning. for example i don't know for sure whether to reduce the problem to this do i really need it to know whether there is more than 1 splitting or not. or on the other hand if i reduce it to the above, is that any kind of improvement at all or not)

(any opinion on my whole approach is more than welcome.)

that is how i came across this question.