# (Riemannian geometry) differentiability of distance function at first conjugate point

Formulation of question: Consider a complete, simply connected Riemannian manifold $$M$$ with Riemannian metric $$d$$. For $$x,y in M$$ that are distinct but close enough to each other, there is a geodesic $$gamma(t), 0 le t le 1$$ such that $$gamma(0)=x$$ and $$gamma(1)=y$$. Assume that $$gamma$$ is the unique minimizing geodesic joining $$x$$ and $$y$$ and that $$y$$ is the first conjugate point of $$x$$ along $$gamma$$. Further, assume that $$y$$ has a suitable normal neighborhood that is contained in $$M$$. Is $$d(x,y)$$ differentiable at $$y$$ when $$x$$ is fixed?

Background: We know that cut loci carry important information on the topology of $$M$$ and information on how the distance function $$d(x,cdot)$$ behaves at a cut point also reveals important information on $$M$$. Further, a lot of techniques in Riemannian geometry utilize the differentiability of $$d(x,cdot)$$, and in statistics, uniqueness of Frechet mean, i.e., $$arg min_{y in M} int_{M} d^2(x,y) mu(dx)$$ for a probability measure $$mu$$ on $$M$$, depends critically on the cut loci of $$d(x,cdot)$$.

Related posts and results: there is a nice post here smoothness of $$d(cdot,cdot)$$ that discusses differentiability of $$d$$ on $$M times M$$, and a classic result on page 108, i.e., Proposition 4.8, in Takashi Sakai’s classic book “Riemannian Geometry”, that says “Suppose that there exist at least two normal minimal geodesics joining” $$x$$ and $$y$$, then $$d(x,cdot)$$ is not differentiable at $$y$$. These discussions confirm the differentiability of $$d(x,y)$$ when $$y$$ is not a cut point to $$x$$ when $$x$$ is fixed.

Recall that a cut point $$z$$ of $$x=gamma(0)$$ is either a first conjugate point to $$x$$ or there are at least two minimal geodesics joining $$x$$ and $$z$$. Recall also that from Theorem 2.1.12 on page 133 of W.A. Klingenberg’s book “Riemannian Geometry”, within any neighborhood of the first conjugate point $$y$$, there exists a point $$w$$ such that there are at least two distinct geodesics that connects $$x$$ and $$w$$. So, if $$d(x,y)$$ with fixed $$x$$ is indeed differentiable at $$y$$, then we also know that $$d(x,cdot)$$ is not differentiable in any neighborhood of $$y$$ except at $$y$$ itself.

However, none of the above discusses whether $$d(x,y)$$ is differentiable at $$y$$ in the setting of the question stated at the beginning. Is the question trivial that its answer is “No”? If the question is so easy, I must have missed something and would like to be enlightened by someone who knows the subject better. Thank you for your attention to my question.

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