1
$\begingroup$

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!

$\endgroup$
2
$\begingroup$

$F$ 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
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.