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Curvature, holonomy and kinematics

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Curvature, holonomy and kinematics

Hi to all,

Working on kinematics of submanifolds - hypersurfaces in Riemannian ambient spaces I've got the result I describe below:

Suppose $(N,\overline{g})$ is a Riemannian manifold with metric tensor $g$ and associated Levi Civita connection $\overline{\nabla}$, let $(M, g)$ be an immersed hypersurface in $N$ via the inclusion $i: M \rightarrow N$, $g = i^{\star}\overline{g}$ and induced Levi Civita connection $\nabla$.
Take a point $p \in M$, identify $i(p)$ with $p$, let $v = v_{n}n$ be the velocity field of the hypersurface moving in $N$ (w.l.o.g assume the tangential component of the velocity vanishes) where $n$ a unit normal field to $M$.
Then if $F(\cdot, t)$ denotes the differential of the motion $\phi (\cdot, t): M \times I \rightarrow N$ ($I$ is a time intervall) then
\begin{eqnarray}
\frac{\partial}{\partial t}|_{t=0}\overline{\nabla}_{F(t)u}n(t) = \overline{\nabla}_{Ju}\delta n + v_{n}\overline{R}(n, Ju)n
\end{eqnarray}
where the last term involving the multiple of the curvature tensor of the ambient manifold is a result of the implementation of the interrelation between the parallel transport on generic Riemannian manifolds and their Riemannian curvature tensor.

Does anybody have an idea if this is nothing but a triviality, if not is there any reference or any correction to be done? The computations,I have repeteadely checked, seem to be correct. Any suggestion, correction, counter example should be helpful.

Thanks


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