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 



Normal vector field associated to deformations of Riemannian submanifolds
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](http://rads.stackoverflow.com/amzn/click/0817637818)


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


