Let $M$ be an Alexandrov space with $sec \geqslant 0$. Let $p$ and $q$ be points of shortest path $\gamma$ in $M$, that are not end point. Then the tangent cone can split as $C_p=L_p \times \mathbb{R}$. Where $L_p=\{x\in C_p;x \perp \gamma\}$.

The second variation says: "For any sequence $\varepsilon_n \to 0$, there is a subsequence $\{\epsilon_n\}\subset \{\varepsilon_n\}$ and the isometry (parallel transportation) $T: L_p \to L_q$ such that $$ |\exp_p(\epsilon_nx)\exp_q(\epsilon_n T(x))|\leqslant |pq|+o(\epsilon^2_n) $$ Where $x\in L_p$ is any vector.

My question: is $|\exp_p(x)\exp_q(T(x))|$ can be controlled by $|pq|?$ I think this is wrong, but I can't think of a counter example.


First note that in positively curved Riemannian manifolds you do have theestimate $$ |\exp_p(\varepsilon\cdot x)\exp_q(\varepsilon\cdot T(x))|\leqslant |pq|. $$ for small $\varepsilon>0$ ONLY. Consider Berger sphere and geodesic along Hopf fiber.

I do not know example of Alexandrov space where this inequality fails for all small $\varepsilon$ but I would not be surprized if one can build it based on the example above.

Yet closely related technical question. In principle the parallel transportation $T$ may depend on the sequence $\varepsilon_n$, but there are no known examples when it does happen. If such examples exist then for fixed $T$ the function $\varepsilon\mapsto|\exp_p(\varepsilon\cdot x)\exp_q(\varepsilon\cdot T(x))|$ might behave quite bad...


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