dependent 3D-metric $g_{t}$ on the slices $B_{\phi \ ,t}$ (see, for example, [@FM]).

To obtain the expression for $g_{t}$ in material coordinates $X^{I}$, we notice that the tangent vectors $$\xi _{I}=-\frac{\phi ^{0}_{,I}}{\phi ^{0}_{,0}}\partial
_{X^{0}}+\partial _{X^{I}},\ I=1,2,3$$ form the basis of the tangent spaces to the slices $B_{\phi ,t}.$ In this basis, $g_{t}$ is given by $$g_{AB}=g(\xi _{A},\xi _{B})=G_{AB}-G_{0B}\frac{\phi
^{0}_{,A}}{\phi ^{0}_{,0}}-G_{A0}\frac{\phi ^{0}_{,B}}{\phi
^{0}_{,0}}+G_{00}\frac{\phi ^{0}_{,A}\phi ^{0}_{,B}}{\phi ^{0\
2}_{,0}},\ A,B=1,2,3.$$

For a synchronized deformation $\gamma = G-(G^{00})^{-1}dT\otimes
dT$ and $g_{t}$ is simply the restriction of 4D-metric $G$ to the slices $B_{T=c}$, i.e. $g_{t\ IJ}=G_{IJ}\vert_{T=t}$, see (2.8).

Denote by $h_{t}$ the 3D metric on the leaves $B_{\phi,t}$ induced by the metric $G_{0}$ (that is by the tensor $\gamma
_{0}=G_{0}-u_{0}\otimes u_{0}$).

Associated with the tensor $\gamma$ there is the projector ((1,1)-tensor) $$\Pi=G^{-1}\gamma =I-u^{*}_{\phi} \otimes u_{\phi}=^{s}\ I-\frac{G^{I0}}{G^{00}}\partial _{X^{I}}\otimes dT=\left(\begin{array}{cc} 0 & 0\\
N^{I} & I_{3}
\end{array}\right) .$$ on the tangent spaces to the slices $B_{\phi ,t}$, last equality being true for synchronized deformations $\phi $.

Let us consider the pullback $\ C_{4}(\phi )=\phi ^{*}\hat{h}$ of the degenerate tensor $\hat{h}$ by the 4D-deformation mapping $\phi $. Tensor $C_{4}(\phi )$ is degenerate in $P$, its kernel is generated by the vector $\phi ^{-1}_{*}(\frac{\partial }{\partial t})$. In coordinates $(X^{I})$ 