  We consider differential operators acting on densities of arbitrary weights
on manifold $M$ identifying pencils of such operators with operators on algebra
of densities of all weights. This algebra can be identified with the special
subalgebra of functions on extended manifold $\hat M$. On one hand there is a
canonical lift of projective structures on $M$ to affine structures on extended
manifold $\hat M$. On the other hand the restriction of algebra of all
functions on extended manifold to this special subalgebra of functions implies
the canonical scalar product. This leads in particular to classification of
second order operators with use of Kaluza-Klein-like mechanisms.
