If I understand correctly, the standard $\mathbb{R}^4$ is diffeomorphic to $\mathbb{R}\times W$ where $W$ is the Whitehead manifold (i.e., is an open three-manifold that is contractible and not diffeomorphic to $\mathbb{R}^3$).

If we consider the standard euclidean metric on $\mathbb{R}^4$ what, if anything, can be said about the (submanifold) geometry of the image of $ \lbrace 0 \rbrace \times W$ under this identification?

For instance, this hypersurface can't be totally geodesic (otherwise it would be diffeomorphic to $\mathbb{R}^3$), but can it be asymptotically flat? Asymptotically conical? Is the (asymptotic) geometry always more complicated? Are there any natural geometric restrictions?