Let $M$ be a compact $C^{\infty}$ Riemannian manifold.Given $p$ and $q$ in $M$ and $T>0$, define $n_{T}(p,q)$ as the number ofgeodesic segments joining $p$ and $q$ with length $\leq T$.Mañé showed in [7] that\[\lim_{T\rightarrow \infty}\frac{1}{T}\log \int_{M\times M}n_{T}(p,q)\,dp\,dq= h_{\rm top},\]where $h_{\rm top}$ denotes the topological entropy of the geodesic flow of$M$.

In this paper we exhibit an open set of metrics on the two-sphere for which\[\limsup_{T\rightarrow\infty}\frac{1}{T}\log n_{T}(p,q)< h_{\rm top},\]for a positive measure set of $(p,q)\in M\times M$.This answers in the negative questions raised by Mañé in[7].