  This paper generalizes an earlier result by the author based on
well-established embedding theorems that connect the classical theory of
relativity to higher-dimensional spacetimes. In particular, an $n$-dimensional
Riemannian space is said to be of embedding class $m$ if $m+n$ is the lowest
dimension of the flat space in which the given space can be embedded. To study
traversable wormholes, we concentrate on spacetimes that can be reduced to
embedding class one by a suitable transformation. It is subsequently shown that
the extra degrees of freedom from the embedding theory provide the basis for a
complete wormhole solution in the sense of obtaining both the redshift and
shape functions.
