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.