# Natural bundle, g-natural metric, meaning

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!

## 1 Answer

$F$ 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