I am trying to understand meaning and importance of a g-natural metric. Since I do pure differential geometry for my research, I am not familiar with many notions which are needed for understanding a g-natural metric. 
My motivation for this question is an example of a Kaehlerian manifold structure which can be constructed on a tangent bundle using a method given by V. Oproiu. His metric is a special kind of a g-natural metric.
Apparently, a g-natural metric is a generalization of Sasaki metric, Cheeger- Gromoll metric etc.


Firstly, we need to understand the notion of an F-tensor fields of type $(r,s)$. 
If $T$ is a tensor field of type $(1,s)$ of a manifold $M$. and $p\_{M}:TM\to M$ the natural projection, $F$ the natural bundle with
\begin{align}
FM=p\_{M}^{\*}(T^{\*}\otimes .. \otimes T^{\*}\otimes T\otimes .. \otimes T)M\to M,
\end{align}
\begin{align}
Ff(X\_{x}, S\_{x}) = (Tf.X\_{x}, (T^{\*}\otimes .. \otimes T^{\*}\otimes T\otimes .. \otimes T)f.S\_{x})
\end{align}
for all manifolds $M$, local diffeomorphisms $f$ of $M$, $X\_{x}\in T\_{x}M$ and $S\_{x}\in (T^{\*}\otimes .. \otimes T^{\*}\otimes T\otimes .. \otimes T)\_{x}M$.
We call the sections of the canonical projection $FM\to M$, $F$-tensor fields of type $(r,s)$.


What is the meaning of $FM$ and $Ff(X\_{x}, S\_{x})$? Is there some example of it? 


Thank you!



Natural bundle, g-natural metric, meaning
$F$ is a functor from the category of smooth manifolds of a fixed dimension whose morphisms are local diffeomorphisms into the category of smooth manifolds, mapping $M$ to a fiber bundle over $M$. All these are associated bundles to a higher order frame bundle for suitable actions of the jet group on a typical fiber. 
See section 14 of:


* Ivan Kolár, Jan Slovák, Peter W. Michor: Natural operations in differential geometry. Springer-Verlag, Berlin, Heidelberg, New York, (1993) [pdf](http://www.mat.univie.ac.at/~michor/kmsbookh.pdf)


