  We contribute to the program of extending computable structure theory to the
realm of metric structures by investigating lowness for isometric isomorphism
of metric structures. We show that lowness for isomorphism coincides with
lowness for isometric isomorphism and with lowness for isometry of metric
spaces. We also examine certain restricted notions of lowness for isometric
isomorphism with respect to fixed computable presentations, and, in this vein,
we obtain classifications of the degrees that are low for isometric isomorphism
with respect to the standard copies of certain Lebesgue spaces.
