TITLE: Developable 3-manifolds in $\mathbb{R}^4$
QUESTION [5 upvotes]: Is there a classification of the equivalent of a "developable surface" in $\mathbb{R}^4$?
Analogous to: planes, cylinders, cones, and tangent developables in $\mathbb{R}^3$?
Edit: Here I am imagining "developing" a 3-dimensional manifold embedded in $\mathbb{R}^4$ into
$\mathbb{R}^3$. (Apologies for the earlier misleading version!)
I would appreciate any suggestions for source materials here.  My only source 
(Edit: now evidently misleading)
is one page (p.342) in Hilbert and Cohn-Vossen (Geometry and the Imagination), in which they say: in $\mathbb{R}^4$

there are surfaces that are isometric to the Euclidean plane in the small but are not ruled.

But now I see from the comments that this must mean a two-dimensional surface embedded
in $\mathbb{R}^4$, which is not exactly what I seek.
A precise definition of developable 3-manifold in $\mathbb{R}^d$ would also be much appreciated.
Thanks for pointers!

REPLY [6 votes]: This is really just an amplification of Deane's answer.  As he points out, it is indeed true that a flat submanifold $M^n\subset \mathbb{R}^{n+1}$ whose second fundamental form is nonvanishing is canonically ruled by $(n{-}1)$-planes.  However, just being ruled in this way does not imply that the submanifold is flat.  For example, the hyperboloid of one sheet in $\mathbb{R}^3$ is ruled by lines (in two ways, in fact), and it is not flat.
The explicit description of the local flat hypersurfaces in $\mathbb{R}^{n+1}$ is, of course, classical, and it goes like this:  Start with a regular curve $p:(a,b)\to S^n\subset\mathbb{R}^{n+1}$, choose a smooth function $g:(a,b)\to\mathbb{R}$, and solve the differential equation $dq = g\ dp$, yielding a smooth curve $q:(a,b)\to\mathbb{R}^{n+1}$.  Now consider the set $M$, consisting of the vectors of the form $q(t) + v$ where $v$ satisfies $v\cdot p(t) = v\cdot p'(t) = 0$.  This will be a (ruled) hypersurface most places, and, where it is, the induced metric will be flat.  Note that this says that such hypersurfaces depend on $n$ functions of 1 variable (the first $n{-}1$ of them come from the curve $p$ and one more comes from the function $g$).