Following a suggestion made by J.-P. Demailly, for each $k\ge
1$, we endow, by an induction process, the $k$-th (anti)tautological
line bundle $\mathcal{O}_{X_{k}}(1)$ of an arbitrary complex
directed manifold $(X,V)$ with a natural smooth Hermitian
metric. Then, we compute recursively the Chern curvature form
for this metric, and we show that it depends (asymptotically---in
a sense to be specified later) only on the curvature of $V$
and on the structure of the fibration $X_{k}\to X$. When $X$
is a surface and $V=T_{X}$, we give explicit formulae to write
down the above curvature as a product of matrices. As an application,
we obtain a new proof of the existence of global invariant
jet differentials vanishing on an ample divisor, for $X$ a
minimal surface of general type whose Chern classes satisfy
certain inequalities, without using a strong vanishing theorem
[1] of Bogomolov.