The manifold discussed here is a smooth manifold equipped with a metric tensor of dimension n, denoted by (M,g). Here I am considering choosing a coordinate ($\psi$,U) on an embedded manifold of codimension 1, i.e. the hypersurface, along with its normal vector field $n^{a}$ which is normalized. Now I can start from any point $p\in U$, find a curve through p with $n^{a}$ a tangent vector and name it $C_{p}$ with parameter $\lambda$, and let $p=C_{p}(0)$. Then I translate every base vector from the surface a little bit along the curve bundle on $U$ and find their new integral curves. Now I have a coordinate for every point slightly above the surface, say $q=C_{p}(\lambda)$, its coordinate is $(\lambda,\psi(p))$. Am I right ? Thanks for judging.
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