For a Nakano semipositive

vector bundle (E, h) on X, we have an isomorphism

We therefore propose to represent them as a

vector bundle < with an added information loop: O.

We can consider PTM as the base manifold of the

vector bundle P*TM, pulled back with the canonical projection map p: PTM [right arrow] M defined by p([x.

The

vector bundle that is constituted by choosing a screen space each point of M is said to be a screen distribution on M, denoted by S(TM).

Let E [right arrow] X be a smooth

vector bundle over an n-manifold X.

2], g) a Riemannian surface endowed with a rank 2

vector bundle E endowed with a metric and a compatible connection [[nabla].

k]M has a multiple

vector bundle structure with k projections onto [T.

0](X), with multiplication given by [E][F] = [E [cross product] F] where E is a

vector bundle and F a coherent sheaf.

1](R,M) is regarded as a

vector bundle over the product manifold R x M, having the fibre type [R.

In [4] Lazarsfeld and the author proved that the polynomials in the Chern classes of a

vector bundle that are positive whenever the bundle is ample are exactly those polynomials that are positive linear combinations of the Schur polynomials.

By definition, the Euler class of the

vector bundle associated to the reduced regular representation [[?

Given a smooth complex

vector bundle [xi], represented by an idempotent matrix [phi] of order n, over the ring of smooth functions on the base, the Chern classes of [xi] in the standard de Rham cohomology of differential forms, are given as follows: