I would like to a find a formula which relates the normal geodesic coordinates associated to a submanifold to the geodesic coordinates on the manifold.
More precisely, let $X$ be a closed submanifold of $Y$. Endow $Y$ with a Riemannian metric $g^{TY}$ and $X$ with the induced metric $g^{TX}$. Denote by $N$ the normal vector bundle of $X$ and induce the natural metric $g^{TN}$ on it so that $g^{TY}|_{X}$ decomposes as $g^{TX} \oplus g^{N}$. Fix a point $x \in X$. By means of the exponential maps $\exp^{X}$ on $X$ and $\exp^{Y}$ on $Y$, define the normal geodesic coordinates $\phi_0 : T_x X \oplus N_x \to Y$ as follows $$ (t, n) \mapsto (\exp_{\exp_x^{X}(t)}^{Y} (n)), \qquad t \in T_x X, \, n \in N, $$ where I am implicitly using the parallel transport on $N$ associated by the projection of Levi-Civita connection onto $N$. Now, one can equally define the geodesic coordinates $\phi_1 : T_x Y \to Y$ at $x \in Y$ by means of the exponential map $\exp^{Y}$. In a small neighbourhood of $0 \in T_x Y$, the maps are related by a diffeomorphism $\psi : T_x X \oplus N_x \to T_x Y$ such that $\phi_0 = \phi_1 \circ \psi$. Of course, we have $\psi'(0) = Id$. Is it possible to find explicit formulas for the higher derivatives of $\psi$ (at least up to order 2)?
I suppose, the calculation can be done using Jacobi fields, but I cannot find it anywhere in the literature. Remark that I am not assuming that my manifold $X$ is a totally geodesic submanifold.