Suppose $M$ is a simply-connected complete Riemannian manifold with bounded sectional curvature $\delta \leq K \leq \Delta < 0$. Let $p_0\in M$. Define the sequence of points $p_1, \ldots, p_n$ by $$p_{k+1} = \exp_{p_{k}}(U_k), \quad U_k\in T_{p_k}M \text{ is some arbitrary vector}, k\geq 0. $$ The goal is to compare (e.g., give an upper bound for) $d(p_0, p_{n+1})$ with the norm of the sum of $U_k$s, that is, with $$ \left\| U_1 + P_2U_2 + \cdots + P_nU_n \right\|, $$ where $P_i$ is the parallel transport from $p_i$ to $p_0$. Is there a systematic way to do this?
My only shot is some technique (Jacobi field estimates) by Buser and Karcher in this paper (6.6, page 105), but I cannot make it work for more than two vectors, as constructing a Jacobi field is not an obvious task. Any ideas on this?
