  It is shown that the determinant line bundle associated to a family of Dirac
operators over a closed partitioned manifold has a canonical Hermitian metric
with compatible connection whose curvature satisfies an additivity formula with
contributions from the families of Dirac operators over the two halves. This
curvature form is the natural differential representative which satisifies the
same splitting principle as the Chern class of the determinant line bundle.
