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.

Lectures on Differential Geometry, amongst others. Probably in Kobayashi-Nomizu I too. $\endgroup$ – Paul Reynolds Mar 1 '15 at 13:22