  We prove that the geometric Bogomolov conjecture for any abelian varieties is
reduced to that for nowhere degenerate abelian varieties with trivial trace. In
particular, the geometric Bogomolov conjecture holds for abelian varieties
whose maximal nowhere degenerate abelian subvariety is isogenous to a constant
abelian variety. To prove the results, we investigate closed subvarieties of
abelian schemes over constant varieties, where constant varieties are varieties
over a function field which can be defined over the constant field of the
function field.
