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Take: $M$ a Riemannian manifold, ${X_0}\in M$, $N_{X_0}$ a submanifold of $M$ going through ${X_0}$, and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.

At ${X_0} \in N_{X_0}$, we consider the orthogonal splitting of the tangent space: $T_{X_0} M=T_{X_0} N_{X_0} \oplus H$. The coordinates of $Z$ can be written $(z,F(z))$. More precisely, we have:

$$F(z)^a=-\frac{1}{2}h(0)_{kl}^a z^kz^l +O(||z||^3)$$

where $h$ is the second fundamental form of $N_{X_0}$ in $M$ at ${X_0}$. $F$ represents the local equation of the submanifold $N_{X_0}$ going through ${X_0}$, in the tangent space at ${X_0}$. Am I right so far?

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Now we assume that there is an isometric group action on $M$. Thus $M$ is foliated by submanifolds $N_X$ (the orbits) which we index by $X$. We choose the points $X$ lying on a geodesic crossing all orbits orthogonally.

We consider a ${X_0}$ and work in the tangent space at ${X_0}$. We take a submanifold $N_X$, going through $X$, a point in a neighborhood of $X_0$. We take $Z \in N_X$ such that $Z$ is in a neighborhood of ${X}$.

My questions are:

Can we say something about the coordinates of $Z$ in $T_{X_0} M=T_{X_0}N_{X_0} \oplus H$? Could we write something like $(z,x+F(z,x))$ where $x$ are the coordinates of $X$? What would be $F$? Can we have a Taylor expansion of $F$ similar to the one above, but in terms of $O(|(z,x)|^n)$?

Is there a theory on Riemannian foliations dealing with these questions? I couldn't find anything but abstract theorems.

Many thanks in advance for your help!

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