Let $M$ be a manifold and let $g$ be a tensor on it, say for example a metric $g\in\Gamma(T^{\ast}M\otimes T^{\ast}M)$. I know how to perform any computation on $g$. For instance, taking its derivative respect to a connection $\nabla$, evaluating it at a point, taking its Lie derivative, obtaining the curvature of the Levi-Civita connection etc.


However, there is a dual formulation on the frame bundle $F(M)$ of $M$, but I never knew how to do the same calculations on the frame bundle, and as I understand it is sometimes simpler to work on the frame bundle. I would like to know how a tensor on $M$ is represented from the point of view of the frame bundle, and how are the typical operations (curvature, Lie derivative etc) implemented. A tensor in $M$ is a section of the corresponding tensor vector bundle. How is this mapped to the frame bundle? 


For example, given an open set $U$ of the atlas of $M$ I can write $g$ in coordinates as follows


$g = g\_{ab}\,dx^{a}\otimes dx^{b}$


What would be the analog local expression from the point of view of the frame bundle?


Finally, I would like to know a reference where these things are explained in detail.


Thanks.



Tensor calculus on the frame bundle
There are some computations in Appendix F of MSM 144 (AMS), following Hamilton's Harnack estimate paper. Under the fair use rule, we've put a link to this at:


www.math.ucsd.edu/~benchow/TensorCalculusFrameBundle.pdf


The material is geared toward an application in Ricci flow.


