Let $\{e_i\}$ be local frame on a Riemannian manifold, constructed as follows: let $\{\hat e_i\}$ be an orthonormal basis of $T_yM$ for some $y\in M$ fixed, and define in a small neighbourhood $U$ of $y$ the vector fields $e_i(x):=P_{\gamma_x}(\hat e_i)(1)$, where $\gamma_x$ is the unique unit speed geodesic from $y$ to $x$, and $P$ denotes the parallel transport system associated to the Levi Civita connection. Let now $\delta(t)$ be a geodesic in $U$. My question is
Is it true that $\frac{d}{dt}\big\vert_{t=0}e_i(\delta(t))$ is horizontal?
This is true by definition of the frame if $\delta=\gamma_x$, and if the connection is flat. But is it still true in the general case? And if not, is there a way to characterise "how far it is from being horizontal"?