Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle
$$TM \vert_{\partial X} = T \partial X \oplus N$$
Where $N$ denotes the normal bundle to $\partial X$. Let $x_0 \in \partial X$ and $U_{\eta, \varepsilon} \simeq B_{\partial X}(x_0, \varepsilon) \times (-\eta, \eta)$ a portion of a tubular neighborhood of $\partial X$ centered at $x_0$ such that $U_{\eta, \varepsilon}$ is contained in a normal coordinate ball $B_M(x_0, \varepsilon’)$.
The adapted Fermi coordinates are given by the map
$$\psi_{x_0}(Z, x) = \exp^M_{\exp^{\partial X}_{x_0}(Z)}(x \mathfrak{n}(\exp^{\partial X}_{x_0}(Z))), \quad Z \in B_{T_{x_0}\partial X}(0, \varepsilon), x \in (-\eta, \eta)$$
where $\mathfrak{n}(x) \in N_x$ denotes the inward pointing unit normal at $x \in \partial X$.
Now let $(E, h^E)$ a hermitian vector bundle with hermitian connection $\nabla^E$ over $X$ and $(e_i)_i$ a local orthonormal basis of $E_{x_0}$, and $(g_i)_i$ the basis obtained by parallel transport of $(e_i)$ for $\nabla^E$ along the curves
$$\gamma_1(t) = \psi(tZ, 0), \qquad \gamma_2(t) = \psi(Z, tx), t \in [0, 1]$$
Let $(f_i)_i$ the basis obtained by radial parallel transport of $(e_i)_i$ for $\nabla^E$ from $x_0$ to the point $\psi(Z, x)$. I am looking for clues, ideas or references on how to compute the first terms of the Taylor expansion of the matrix $P(\psi(Z, x))$ defined by
$$f_i = \sum_k P_{ik}(\psi(Z, x)) g_k$$
The first term is the identity (value of $P(\psi(Z, x))$ for $Z=0$ and $x=0$). What I am interested in is a way to find the second term.
Thank you for your help.
Edit : The main reference I am aware of regarding approximation between normal and Fermi coordinates is Alfred Gray’s Tubes, Chapter 9 where the author works with the Levi-Civita connection on the tangent bundle, but I am not confident the method adapts to the case of any vector bundle.
