Let
be a submanifold of dimension
of the complex projective space
(
), and let
be a vector bundle of rank two on
. If
we prove a geometric criterion for the existence of an
extension of
to a vector bundle on the first order infinitesimal
neighborhood of
in
in terms of the splitting of the
normal bundle sequence of
, where
is the zero locus of a general section of a high twist of
. In the last section we show that
the universal quotient vector bundle on the Grassmann
variety
of
-dimensional linear subspaces of
, with
and
(i.e. with
not a projective space), embedded in any projective
space
, does not extend to the first infinitesimal neighborhood of
in
as a vector bundle.