I'm interested in the derivation of the Gauss equation (Gauss-Codazzi). Usually we consider the definition of the Riemann tensor on the hypersurface.
$$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c$$
where $D$ is the connexion associated to the induced metric ($h$) on the hypersurface.
We have
\begin{align} D_aD_bw_c=D_a\Bigl[h_b^{~d} h_c^{~e}\nabla_d w_e\Bigr]=h_a^{~f}h_b^{~g}h_c^{~e}\nabla_f\Bigl[h_g^{~d}h_k^{~e}\nabla_dw_e\Bigr] \end{align}
where $\nabla$ is the connexion of the metric ($g$) in the full space.
Hence after some algebra we can arrive at the desired result. What I don't understand is why can we not do:
$$ D_aD_bw_c=D_a\Bigl[h_b^{~d} h_c^{~e}\nabla_d w_e\Bigr]=h_b^{~d} h_c^{~e}D_a\Bigl[\nabla_d w_e\Bigr] $$
because $D_a h^b_c=0$.
But in that case I would have
$$ D_aD_bw_c=h_b^{~d} h_c^{~e}D_a\Bigl[\nabla_d w_e\Bigr]=h_b^{~d} h_c^{~e}h_a^{~\mu}h_d^{~\nu}h_e^{~\sigma}\nabla_\mu\Bigl[\nabla_\nu w_\sigma\Bigr]=h_a^{~\mu}h_b^{~\nu}h_c^{~\sigma} \nabla_\mu\nabla_\nu w_\sigma $$
Therefore we would have
$$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c=h_a^{~\mu}h_b^{~\nu}h_c^{~\sigma} ~^{(n)}R_{\mu\nu\sigma}^{~~~~~~~~d}w_d$$
which is not the desired result.
So I understand the standard derivation, but I don't understand why in the way that I wrote, it doesn't work ?
