  Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of
pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show
that the Forni bundle of N (i.e. the maximal SL(2,R)-invariant isometric
subbundle of the Hodge bundle of N) is always flat and is always orthogonal to
the tangent space of N. As a corollary, it follows that the Hodge bundle of N
is semisimple.
