Let $(M,g)$ be a $d$-dimensional Riemannian manifold and $\pi: O(M)\to M$ be the corresonding orthonormal frame bundle. Denote by $d$ the distance function on $M$ induced by the Riemannian metric $g$. For $\xi\in\mathbf R^d$, denote by $H_\xi$ the standard horizontal vector field corresponding to $\xi$. Let $\gamma$ be a smooth curve on $\mathbf R^d$. Consider the following ODE on $O(M)$: \begin{equation} \dot u_t = H_{\dot\gamma_t}(u_t). \end{equation} This is the so called Cartan's development. Define $x$ to be the projection curve onto $M$ of the solution $u$, i.e., $x_t = \pi(u_t)$. Then $$\dot x_t = \pi(\dot u_t) = u_t(\dot\gamma_t).$$ Since each mapping $u_t: \mathbf R^d \to T_{x_t}(M)$ is isomorphic, we have $$\int_0^t g(\dot x_s,\dot x_s)^{1/2} dt = \int_0^t |\dot\gamma_s| ds.$$ That is, the curve $x$ on $M$ and the curve $\gamma$ on $\mathbf R^d$ have the same length. But my question is:
How to compare the two quantities $d(x_t,x_0)$ and $|\gamma_t-\gamma_0|$?
I guess that this must relate to the curvature of $M$. But I do not know how to derive. Any or hints or reference will be appreciated. TIA.
EDIT: Assume $M$ is geodesically complete so that the equation of Cartan's development can be globally solved.