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Space and times derivatives, inevrtibility

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Space and times derivatives, inevrtibility

Hi to all,

take a Riemannian manifold $(\overline{M}, \overline{g})$ and a time dependent family of smooth embeddings $\phi( \cdot , t)$ of a hypersurface $(M,g)$ immersed via the inclusion mapoing $j: M \rightarrow \overline{M}$. For each $p \in M$ denote by $F_{t}(p): T_{p}M \rightarrow T_{\phi_{t}(p)}\overline{M}$ the differential of $\phi(, t)$.
We know that in the case of an Euclidean ambient space we have no problem to invert the order of differentiation between space and time derivatives.
I guess that this is not in general truth, it seems that the curvature of the ambient space should involve in the situation.
Can anyone help with this or the invertibility is valid no matter the ambient manifold is a curves one or not?

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