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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?

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  • $\begingroup$ Only a vague idea: have you tried the Toponogov comparison theorem? This and many others are covered in a book by Cheeger and Ebin. Maybe you find something helpful there. Of course, that theorem would at first help only with two vectors, because you would need a good estimate on $\exp^{-1}_{p_0}(p_2)$ to continue. But my impression is that global comparison results might be more helpful than Jacobi field comparison. $\endgroup$
    – Sebastian Goette
    Nov 18, 2021 at 12:25
  • $\begingroup$ Very good idea indeed! However, I was thinking that with Topogonov, I might be able to transfer the problem to a space with constant sectional curvature $\delta$. It is still unclear for me how to deal with this problem even in that case... (scratching my head) $\endgroup$
    – M.R.Karimi
    Nov 18, 2021 at 19:55
  • $\begingroup$ In a space of constant curvature, there are trigonometric formulas that you can apply (for the first two vectors). Then you have to "guess" the next angle and continue ... $\endgroup$
    – Sebastian Goette
    Nov 18, 2021 at 22:01

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