I had a couple of related questions on the cut locus, conjugate points and smoothness of distance function. Let $(M,g)$ be a smooth complete Riemannian manifold and $r(x) = d(p,x)$ the distance function to a fixed point $p\in M$. Recall that the cut locus $Cut(p)$ consists of two kinds of points - points that are conjugate to $p$ along some geodesic, or points that have multiple minimal geodesics connecting them to $p$.

1) I know that $r$ is smooth on $M\backslash (Cut(p)\cup \{p\}$. Is it true that $r$ is in fact smooth at $x$ as long as there is a unique minimal geodesic connecting it to $p$? The only proof of smoothness involves the exponential map, but the exponential map is not invertible at conjugate points, so that approach wont work.

2) Is there a simple example of $(M,g,p)$ with points conjugate to $p$ but with unique minimal geodesics to $p$.


1 Answer 1

  1. The distance function has to be differentiable at $x$ if there is a unique minimizing geodesic $[p,x]$. It means that the differential $d_xr$ is well defined linear function on the tangent space $\mathrm{T}_x$. The proof is straightforward. But, it does not mean that $r$ is smooth in a neighborhood of $x$ --- evidently it does not hold if $x$ is conjugate to $p$ along the geodesic.

  2. Take a surface of revolution for an even function $f$; say $f(0)=1$, $f''<0$. You can make it so that the curvature on the equator $(f(0)\cdot \sin t,f(0)\cdot\cos t,0)$ is positive and it takes maximal value on the surface. Then the maximal minimizing arc on the equator is unique minimal geodesics between its ends.

  • $\begingroup$ I'm sorry to disturb six years later, but I really don't see why 1) is true. That there is a unique minimizing geodesic from $p$ to $x$ does not mean the same is true in a neighborhood of $x$, so we can't apply the first variation formula. So how do we prove the distance function is differentiable? $\endgroup$
    – Yuxiao Xie
    Feb 9, 2020 at 10:32
  • $\begingroup$ @Colescu If the function is not differentiable at $x$, then there is a sequence $x_n\to x$ such that $\measuredangle[x_p^{x_n}]+\measuredangle[{x_n}_p^{x}]>\pi+\varepsilon$ for some fixed $\varepsilon>0$. Observe that the geodesics $[p, x_n]$ do not converge to $[p,x]$, whence its limit is another geodesic from $p$ to $x$. $\endgroup$ Feb 10, 2020 at 0:04
  • $\begingroup$ What do you mean by ∡[xxnp]+∡[xnxp]? (sorry, don't even know the latex commands) $\endgroup$
    – Mathy
    Feb 12, 2020 at 7:23
  • $\begingroup$ @Mathy angles. $ $ $ $ $ $ $ $ $\endgroup$ Feb 13, 2020 at 6:30
  • $\begingroup$ Yeah, what's the angle? I don't get your notation, an angle is something between 2 things, and you just out one argument. $\endgroup$
    – Mathy
    Feb 13, 2020 at 6:37

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