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.



Number of geodesics of certain length
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 $i$.


