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Let $(M,g)$ be a Riemannian manifold of dimension $n$ and $X$ be a immersed submanifold in $M$ of dimension $k$ i.e there is a immersion $F_{0}:X \longrightarrow M$.

A deformation of the submanifold $X$ is defined by a smooth family of immersions $F : I \times X \longrightarrow M$ i.e. $F_{t}: X \longrightarrow M$ is an immersion for all $t \in I$ and $F_{0}$ is the immersion of $X$ defined above, so we have a family of immersed submanifolds $F(t,(X))=X_{t}$.

Let $F_{*}( \frac{\partial}{\partial t})$ be the deformation vector field on $M$ associated to this deformation i.e. for any $p\in X$ take the curve $F(\cdot,p):I \longrightarrow M$ and $F_{*}( \frac{\partial}{\partial t})$ at $p$ is the tangent vector to this curve at $t=0$.

Now this is my problem: I have read that if the submanifold $X$ is compact and orientable then we can find a family of diffeomorphisms of $X$ depending on $t$ such that we can assume that the vector field of the deformation is normal to $X_{t}$ for all $t$. It is not clear how this reparametrization is done, if I consider the submanifold $X_{0}$ and the deformation vector field on it, i has 2 components corresponding to the splitting $TM \vert_{X}=T(X) \oplus N(X)$ how I get rid of the tangent component using diffeomorphisms of $X$ as it is stated above?

Thanks for your help

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    $\begingroup$ Project $\tfrac\partial{\partial t}$ to the normal bundle and integrate. $\endgroup$ – Anton Petrunin Feb 10 '14 at 19:56
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If you're looking for an explicit example, see page 8 in Regularity Theory for Mean Curvature Flow by Ecker. The preview on amazon includes the page.

http://www.amazon.com/Regularity-Theory-Mean-Curvature-Flow/dp/0817637818

The equation (2.1) referred to is the mean curvature flow equation, $$\frac{\partial x}{\partial t} = \vec H(x)$$

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  • $\begingroup$ Thanks, the book you mentioned seems to be a good reference in general, very clear explanation how to reparametrize by projecting to the normal vector field (as commented by Anton and Valeri above) $\endgroup$ – Coffee Feb 11 '14 at 2:04
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This is not exactly what you are asking, but in algebraic geometry we would say the following: the family of immersions $(F_t)$ is not uniquely defined, we can replace $F_t$ by $F_t\circ\varphi _t$, where $(\varphi _t)$ is a one-parameter family of diffeomeorphisms of $X$ -- this will give the same family of submanifolds. Thus $F_*(\frac{\partial}{\partial t} )$ is well defined only in the quotient $TM_{|X}/TX$, which is canonically isomorphic to the normal bundle.

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    $\begingroup$ add to abx answer: i.e., just take $\phi_t$ be the one-parameter group of diffeomorphisms of M generated by the tangent component of the deformation field. $\endgroup$ – valeri Feb 10 '14 at 18:17

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