Number of geodesics of certain length

Question

Let $M$ be a Riemannian manifold, and let $x, y \in M$ be non-conjugate points.

Let $r, R>0$ be two numbers. I am looking for a bound on the number of geodesics between $x$ and $y$ of Length between $r$ and $R$, like "there are at most $N(r, R)$ geodesics between $x$ and $y$", depending on the curvature of $M$.

It seems that one needs assumptions on the curvature to obtain such bounds. For example, one might want to assume that that sectional curvature is bounded from below by some $\kappa < 0$.

\Edit: The post of Robert Bryant suggested that one should also assume that $M$ has positive injectivity radius, or that $M$ is compact.

Answer 1

I think that no such bounds exist. For example, you can take $M$ to be a (complete) surface of revolution in Euclidean $3$-space that has negative curvature bounded from below by $-1$ and that is asymptotic to the classic pseudospherical surface (i.e., a 'spike' going off to infinity). On such a surface, you can find a sequence of pairs of nonconjugate points $(x_i,y_i)$ that are very close together while the $x_i$ go off to infinity along the spike and are such that the number of geodesics from $x_i$ to $y_i$ of length between $r$ and $R$ is at least $i$.

Answer 2

I think that for any manifold with negative curvature having positive injectivity radius $\delta>0$ you get an upper bound of the form $C\delta^{-n} e^{c(R+d(x,y))}$ where $n$ is the dimension $c,C$ depend on the curvature and the dimension. The argument I have in mind actually yields a bound depending only on the injectivity radius at $x$ (or $y$) and goes as follows: if you fix one geodesic arc from $x$ to $y$ (say one of the shortest ones), and fix a lift $\tilde x$ of $x$ to the universal cover, then any other arc of length less than $R$ gives you an element in the fundamental group which displaces $\tilde x$ of less than $R+d(x,y)$, and two distinct arcs yield distinct elements. Now there is a well-known bound for the number of translates of $\tilde x$ by elements of the fundamental group at a distance $\le d$ of $\tilde x$ which is of the form $C{\rm inj}(x)^{-n}e^{cd}$ (where $c,C$ come from the asymptotics for the volume of balls in the universal cover), which in turn yields the claimed bound for the number of arcs. (Note that the lower bound $r$ does not play a role if the growth is indeed exponential).