# What sort of geometry does the Whitehead manifold have as a hypersurface in $\mathbb{R}^4$?

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?

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Note that to prove that $W$ is not $\mathbb{R}^3$, one proves that it is not simply connected at infinity in some sense; that should impose condition on the possible asymptotic geometries. –  Benoît Kloeckner Jun 5 '13 at 8:10
Okay so that should imply that it is not asymptotically flat. Good to know –  Rbega Jun 5 '13 at 8:16