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With negative extrinsic curvature convention, the background trace is $\bar{K}=-3H$. The scalar expansion perturbation is $\kappa =K+3H$. For lapse function $N=\bar{N}(1+\Psi )$, induced metric $h_{ij}=a^2[(1-2\Phi ){\delta }_{ij}+2E_{,ij}]$ and shift vector $N_i=-a^2{B}_{,i}$, it is $\kappa =3(\dot{\Phi }/\bar{N}+H\Psi )-\Delta \chi$, where $\chi$ is the scalar shear potential. The expansion of the future normal congruence is $-K$, so its perturbation has the opposite sign.

With signature $+---$ and $n^{\mu }{n}_{\mu }=1$, the spatial projection tensor is $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }-n^{\mu }{n}_{\nu }$. Projecting the spacetime metric tensor gives $q_{\mu \nu }=g_{\mu \nu }-n_{\mu }{n}_{\nu }$, which is negative definite on spatial vectors. The positive spatial induced metric is $h_{\mu \nu }=-q_{\mu \nu }$. A plus sign in $g_{\mu \nu }+n_{\mu }{n}_{\nu }$ belongs to a different normal/signature convention and is not transverse in this one. Fully projected covariant derivatives of either spatial metric vanish by metric compatibility.