An elementary question about the cut locus

Question

Let $M$ be a Riemannian manifold, $x$ and $y$ are two points in $M$. Assume that $x$ is not in the cut locus of $y$. Does there exist a neighborhood $U$ of $x$ and a neighborhood $V$ of $y$ such that for every point $u$ in $U$ and for every point $v$ in $V$ we have that $u$ is not in the cut locus of $v$?

Answer 1

Yes.

The cut locus for a point $x$ is the closure of the set of points $y$ such that there is more than one minimum length geodesic from $x$ to $y$. Sometimes $y$ may be on the cut locus yet have a unique minimizing geodesic to $x$. This happens when all pairs of minimizing geodesics from $x$ to points $y'$ near $y$ converge to give a single minimizing geodesic from $x$ to $y$ in the limit.

In such a case, $y$ is on the conjugate locus for $x$. This situation can be detected by the first derivative of the exponential map at $Y$, which is singular. An equivalent way to phrase it is that there is a nontrivial Jacobi field along the given geodesic from $x$ to $y$ that is 0 at $x$ and at $y$. (A Jacobi field is the first derivative in the parameter direction of a 1-parameter family of geodesics, that is, an infinitesimal variation of a geodesic). Note that this condition is symmetric in $x$ and $y$.

When $y$ is not a conjugate point of $x$ so that there is no Jacobi field of this sort, then by the implicit function theorem there is a smooth family of geodesics parametrized by $U \times V$ connecting points in a neighborhood $U$ of $x$ to points in a neighborhood $V$ of $y$.

For a complete Riemannian manifold $M$, the set of cut pairs $C(M) \subset M \times M$ is a closed subset invariant under interchanging factors: for any pair of points $(s,t)$ that is a limit of pairs of points $(s', t')$ joined by more than one minimizing geoesic, either there are distinct limits of sequences of minimizing geodesics, giving more than one minimizing geodesic from $s$ to $t$, or $s$ and $t$ are a conjugate pair. This answers your question yes.

Answer 2

For a unit tangent vector $u$ with footpoint $p$ let $t(u)$ be the supremum of positive numbers such that the geodesic $t\to \exp_p(tu)$ is minimizing on $[0,t(u)]$. The cut locus at $p$ is the set of points $\exp_{p}(t(u) u)$ of $M$ for which $t_u$ is finite.

A basic result is that $u\to t(u)$ defines a continuous map from the unit tangent bundle to $(0,+\infty]$ where continuity at $+\infty$ is understood in the obvious way. See e.g. Sakai's "Riemannian geometry", Proposition III.4.1.

Now coming to your question fix $x\in M$ and $y=\exp_x(su)$ for some $u=u(x,y)$ and positive number $s$. Suppose $x^\prime$, $y^\prime$ are near $x$, $y$ respectively, and write $y^\prime=\exp_{x^\prime}(s^\prime u^\prime)$. If $y$ is not in the cut locus of $x$, then $t(u)=+\infty$. So $t(v)> s$ on some neighborhood of $u$ in the unit tangent bundle. Since $s^\prime$ and $s$ are almost the same, we conclude that $t(u^\prime)> s^\prime$, i.e. $x^\prime$, $y^\prime$ are not cut points of each other.

In my view Sakai's book is the best source of information about cut and cunjugate loci.