Similarly, introduce the isometric embedding
[R.sub.0,n] ([L.sub.2][(R)).sup.n] [right arrow] [L.sub.2](R) 8i2(R) by the rule
If G is an open subgroup of invertible elements of A and T : [Gx.sub.0] [right arrow] Y is an isometric embedding, where Y is either [C.sub.R](K) for some compact Hausdorff space K or is a strictly convex real Banach space, then there exists [y.sub.0] [member of] Y and a real-linear isometry [??] : [Ax.sub.0] [right arrow] Y such that [??]([ax.sub.0]) = T([ax.sub.0]) - [y.sub.0], for all a [member of] G.
Let T : U [right arrow] Y be an isometric embedding. Then there exists a surjective linear map S : Y [right arrow] span U such that S(T(u)) = u for all u [member of] U and the restriction of S to span T(U) has norm one.
Any isometric embedding of G with idim(G) into [Q.sub.h] describes the same family of semicubes and pairs of complementary semicubes, which are indexed in a different way.
For an edge ab of a partial cube with an isometric embedding [alpha] into [Q.sub.h], let the vertices a and b differ in the coordinate i, i.e.
Of course, abstract infinite dimensional manifolds appear intimidating and it is only natural to seek a "reassuring" result, namely an infinite dimensional counterpart, if not of Nash's Isometric Embedding Theorem, then at least of Whitney's Topological Theorem (see, e.g.
Saucan, Isometric Embeddings in Imaging and Vision: Facts and Fiction, to appear in J.
Here we study spacetimes admitting local and isometric embedding into [E.sub.6], i.e.,4-spaces of class two [1,2].
In this work we study spacetimes (here denoted as [V.sub.4]) admitting a local and isometric embedding into the pseudo-Euclidian [gamma]at space, that is, spacetimes of class two [1,2].
This is a systematic presentation of results concerning the isometric embedding
of Riemannian manifolds in local and global Euclidean spaces, especially focused on the isometric embedding
of surfaces in a Euclidean space R3 and primarily employing partial differential equation techniques for proving results.
By Lemma 3.3, there is an isometric embedding
F: A [right arrow] [E.sup.n+2], F([x.sub.0]) = [x.sup.*.sub.0], ..., F([x.sub.n+2]) = [x.sup.*.sub.n+2].
Much later, when studying how U may be embedded into the Banach space C[0,1] of all continuous functions from [0,1] to R equipped with the sup norm , Holmes discovered that U has the following remarkable property: for every isometric embedding