Let $P$ be a point on compact Riemannian manifold $M$. Let $L$ be the cut locus. Let $Q$ be a smooth point of $L$. Is $L$ totally geodesic in a neighborhood of $Q$?
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The cut locus is in general not a submanifold. How do you define "totally geodesic"? – Yuxiao Xie Jan 28 '20 at 02:37