= Solution
The <unimodular spatial metric> separates volume from shape: writing $h_{ij}=a^2\widetilde h_{ij}$ and $a^6=h=\det h_{ij}$ gives $\det\widetilde h=1$. Contract the <extrinsic curvature with a negative shift> with $h^{ij}$. <Metric compatibility> and the <Jacobi determinant derivative formula> give
$$
h^{ij}\dot h_{ij}=\partial_t\log h,\qquad h^{ij}(D_iN_j+D_jN_i)=2D_iN^i.
$$
Consequently
$$
\boxed{K=-\frac1{2N}\left(\frac{\dot h}{h}+2D_iN^i\right).}
$$
For zero <shift vector>, $\dot h/h=6\dot a/a$, so
$$
\boxed{H=-K/3=\dot a/(Na).}
$$
The infinitesimal spatial volume is $\sqrt h\,d^3x=a^3d^3x$; its logarithmic change per unit normal <proper time> is $3H$. Thus this $H$ is the <local volume Hubble parameter>, reducing to the usual <Hubble parameter> in a homogeneous universe. The determinant-defined $a$ refers to fixed spatial coordinates, while this normal volume expansion has a geometric interpretation.
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