I am currently trying to understand the paper "Global Wave Maps on Robertson–Walker Spacetimes" by YVONNE CHOQUET-BRUHAT and don't understand the following part
where this $D$ is defined. The context is the following: We have a Lorentzian manifold $(V, g)$ where $V=S \times R$, $S$ is an $n$-dimensional smooth manifold. $g$ is a Robertson-Walker metric. Then we have a Riemannian manifold $(M,h)$.
I don't understand what "the covariant derivative in the metrics $\sigma$ and pullback of $h$ of a mapping from $S$ into tensor products of $TS$ and $TM$ or their duals" is supposed to mean.
My thoughts:
$\nabla: \Gamma(TS) \times \Gamma(\bigwedge ^2 T^{*}S) \rightarrow \Gamma(\bigwedge ^2 T^{*}S)$ so $\nabla \sigma : \Gamma(TS) \rightarrow \Gamma(\bigwedge ^2 T^{*}S)$
Now $h \in \Gamma(\bigwedge^2 T^{*}M)$. Let $f: S \rightarrow TM \otimes TS$. What is the pullback of $h$ of $f$?
I see that it's pretty hard to help here since one needs to read in the paper to understand more but I would appreciate any hint since I have no idea where to start here.