Let $\phi: (M,g)\hookrightarrow (N,\tilde{g})$ be an isometric embedding of a Riemannian manifold $M$ of dimension $m$ into a Riemannian manifold $N$ of dimension $n$. I am interested in trying to do a Taylor expansion of this mapping in Riemannian Normal coordinates.

I have shown previously that for any real-valued function $f: M\rightarrow \mathbb{R}$, the covariant $k$-th derivative $$ \nabla^{k+1} f [X_1,...,X_{k+1}]= X_{k+1}(\nabla^{k}[X_1,...,X_k])-\sum_{i=1}^k\nabla^k f[X_1,...,X_{i-1},\nabla_{X_{k+1}}X_i,...,X_k] $$

written Riemannian normal coordinates (or in in any coordinates where the Christoffel symbols vanish) is simply the expression $$ \sum_{I}X^I \frac{\partial^I f}{\partial X^I} $$ Where $I$ is a multiindex of size $k$ varying over all choices of $m$ integers from the set $\{1,...,m\}$ (This looks like the $k$-th derivative.)

This suggests that when I choose normal coordinates in $M$, and perform a Taylor expansion of $f$, I am actually computing the coordinate expression of $$ \nabla f[X] + \frac{1}{2!}\nabla^2 f[X,X] + \frac{1}{3!}\nabla^3f [X,X,X] + ... $$

If I wish to more generally understand the taylor expansion of a mapping between manifolds, $\phi: (M,g)\hookrightarrow (N,\tilde{g})$ as above, I can choose a coordinate mapping $\eta:U\rightarrow \mathbb{R}^n$ in $N$ and regard the $i$-th component function of the mapping $\eta\circ\phi$.

This suggests to me that there might be a way to Taylor expand $\phi$ in terms of it's covariant derivatives that is independent of choosing a particular chart $\eta$ in $N$. Is this the case? I suppose that the question more generally is is there a way to view a linear connection $\nabla_X Y$ as some combination of covariant derivatives on scalar functions?