Sorry in advance for the long setup:
Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold, $R$ is 1-dimensional. $M$ is $d$-dimensional. $G$ is a Robertson-Walker metric $g=-dt^2+R^2(t)\sigma$. Now i $\partial_i u^A$ and $\partial_t u^A$ are mappings from $S \times R$ into $T^∗ V\otimes TM$.
$u$ is supposed to be a wave map, so $g.\nabla^2 u=0$.
I also have the formula
$(\nabla_{\alpha} \nabla_{\beta}- \nabla_{\beta} \nabla_{\alpha})\partial_{\lambda} u^A =R^{\mu}_{\alpha \beta \lambda}\partial_{\mu} u^A + \partial_{\alpha} u^C \partial_{\beta} u^B R^A_{CBD} \partial_{\mu}u^D$
Here, latin indices $\in \{0,...,n\}$, small letters $i,j,.. \in \{1,..,n\}$ and capital letter indices $A,B,... \in \{1,...,d\}$. $R$ are the Riemann curvature tensors resp. for $V,M$.
Also, $D_i$, respectively $ D_t$ are defined as covariant derivative in the metrics $\sigma$ and pull back of $h$ of a mapping from $S$ into $T^∗ S\otimes TM$;the covariant derivative in the pull back of $h$ of a mapping from $R$ into $TM$. With this definition, the wave equation is the same as
$-R^{-n} D_t(R^n \partial_t u)+R^{-2} \sigma^{ij} D_i D_ju=0$
Now I read the following:
The expression of $\nabla^{\lambda} \nabla_{\lambda} u^A$, the definitions of $D$ and $D_t$ with the use of the Ricci formula for the commutation of covariant derivatives lead to the equations where latin indices are raised with $\sigma$:
$D_i \nabla^{\lambda} \nabla_{\lambda} u^A =-D_tD_i \partial_t u^A -n R^{-1} R^{'} D_i \partial_t u^A+ R^{-2} D^hD_h \partial_i u^A - f^A_i=0$
where
$f^A_i= R^A_{B C D} \partial_i u^B \partial_t u^C \partial_t u^D+R^{-2}( - \theta^j_i \partial_j u^A+ R^A_{ BCD} \partial_i u^B \partial^h u^C \partial_h u^D=0)$
where $\theta_{ij}$ is the Ricci tensor of $\sigma$.
I don't understand this at all. It would already help me a lot if someone could answer me the questions:
- What does "where latin indices are raised with $\sigma$" mean?
- What does the expression $D^hD_h$ mean?
- Why are the expressions $=0$?