= Metric compatibility of the spatial covariant derivative
{title2=$D_\lambda h_{\mu\nu}=0$}
For a unit timelike normal $n$, the <induced metric> is $h_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$. The <spatial covariant derivative> projects the derivative and all free indices with the <spatial projection tensor> $P^\mu{}_\nu=\delta^\mu{}_\nu+n^\mu n_\nu$. <Metric compatibility> gives $\nabla_\lambda h_{\mu\nu}=(\nabla_\lambda n_\mu)n_\nu+n_\mu\nabla_\lambda n_\nu$; projecting kills the normal factor in each term. Thus the <spatial covariant derivative> is compatible with the <induced metric>.
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