Let $\pi:M\to X$ be a fibration map between two spin manifolds, i.e. the fiber $\pi^{-1}(x)$ is a manifold, suppose $s:X\to M$ is an embedding. Let $\Phi$ be a solution of Dirac equation, i.e. $D^X\Phi=0$(we use the Levi-Civita connection to define the spinor bundle). If we have a splitting, $TM=T^HM\oplus T^vM$, here $T^HM\cong \pi^*TX$, given a metric $g^M$, we also define the Dirac operator $D^M$.
Q If $g^M=\pi^*g^X\oplus g^{T^vM}$, and $T^HM=\pi^*TX$, can we say $D^M\pi^*\Phi$=0?
PS: By the calculation, we know that $D^M=D^{M/X}+D^{T^HM}+\sum^{\dim X}_{i=1} div^V(f_i)-\frac{c(T^H)}8$, here $T^H$ denotes the Torsion of $\pi^*\nabla^X\oplus\nabla^{M/X}$ along the horizontal direction, and $div^v(f_i)=\frac12 Tr[(g^{T^vM})^{-1}L_{f_i}g^{T^vM}]$.
I do not if there is any result or estimate about the torsion of connection or divergent.
