  We systematically study the splitting of vector bundles on a smooth,
projective variety, whose restriction to the zero locus of a regular section of
an ample vector bundle splits.
  First, we find ampleness and genericity conditions which ensure that the
splitting of the vector bundle along the subvariety implies its global
splitting. Second, we obtain a simple splitting criterion for vector bundles on
the Grassmannian and on partial flag varieties.
