$\xi_{F}$ of the fibers $$\xi_{F}:((X^{0}=T;X^{I},I=1,2,3) \longmapsto
    (F(T,;X^{J},J=1,2,3),X^{I},I=1,2,3))$$ with $F_{,0}>0.$

2.  For any admissible history of deformations $\phi$ one can choose a transformation $\xi_{F}\in TC$ such that the history of deformation $\phi \circ \xi_{F}$ is synchronized.

To prove the second statement let $\phi $ be an admissible history of deformation. Define the element $F \in TC(P)$ as follows: $F:(T,X^{I},I=1,2,3) \longmapsto (\phi
^{0}(T,X^{J},J=1,2,3),X^{I},I=1,2,3)$. Then, $\phi =\phi _{1}\circ
F$ where $\phi _{1}$ is another admissible history of deformation with the same components $\phi ^{i},i=1,2,3$ and the identity component $\phi _{1}^{0}(T,X^{I},I=1,2,3)=T$. Transformation $F$ is admissible since $F^{0}_{,0}=\phi ^{0}_{,0} >0$, thus $F \in TC(P).$ Apparently, the deformation history $\phi _{1}$ is synchronized.

Therefore we may restrict our consideration to the synchronized histories of deformation keeping in mind that the covariance group of the theory reduces from the group $Aut^{+}(P)$ to the group $D(B)$ of time-independent orientation preserving diffeomorphisms of base $B$.

ADM-decomposition of Material Metric, Lapse and Shift.
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Slicing $B_{\phi ,t}$, generates the (3,1)-decomposition of the material metric $G$ employed (for a Lorentz type metric) in General Relativity ([@MTW],[@ADM]). Specifically, the sandwich structure of the part of the manifold $P$ bounded by the surfaces $B_{\phi ,t}$ and $B_{\phi ,t+\Delta t}$ allows one to introduce 