  In this work we study a large class of exact Lie bialgebras arising from
noncommutative deformations of Poisson-Lie groups endowed with a left invariant
Riemannian metric. We call these structures \emph{exact metaflat Lie
bialgebras}. We give a complete classification of these structures. We show
that given the metaflatness geometrical condition, these exact bialgebra
structures arise necessarily from a nontrivial solution of the classical
Yang-Baxter equation. Moreover, the dual Lie bialgebra is also flat and
metaflat constituting an important kind of symmetry.
