If $c$ is a normal geodesic and if $e_1$ is a unit parallel vector field, then assume that for unit vector field $v,\ v\perp e_1$, $$ R(e_1,v,v,e_1)(t) \leq R(e_3,e_4,e_4,e_3)(t) \ \ast$$ for any unit vector field $e_3,\ e_4$ s.t. $e_3\perp e_4$.

Then we can find parallel vector field of $v$ satisfying $\ast$.

This is true ? (Motivation : (1) If $M$ has nonegative sectional curvature and if $M$ has two disjoint totally geodesic submanifolds $N_i$ s.t. ${\rm dim} N_1+{\rm dim} N_2\geq {\rm dim}\ M$, we have a flat strip between them

(2) On $\mathbb{C}P^2$, recall "canonical" $S^1$-action whose fixed point set is $S^2\cup \{p\}$ Then the above happen along a minimizing geodesic between $S^2$ and $p$. )

Thank you in anticipation.


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