Suppose $g\_1$, and $g\_2$ are two Riemannian metrics on a closed surface $S$, provided that the Gaussian curvature $K\_{g\_1}$ $<$ $K\_{g\_2}\leq -1$. Denote by $\mathcal{C}$ the set of free homotopy classes of closed curves in $S$. The marked length spectrum function is $l\_{g\_i}: \mathcal{C} → \mathbb{R}^{+}$ which assigns to the class $[\gamma]$ the length $l\_{g\_i}(\gamma)$ of the closed geodesic in $[\gamma]$.


My question is that given the curvature condition $K\_{g\_1}$ $<$ $K\_{g\_2}\leq -1$, can one conclude that $$l\_{g1}([\gamma])\leq l\_{g\_2}([\gamma]),$$ for all $[\gamma]\in\mathcal {C}$? i.e. the length of a closed geodesic in $(S,g\_2)$ is longer then the length of the corresponding closed geodesic in $(S,g\_1)$. 


**p.s. the converse is not true. Thanks to the answer from @ Igor Rivin and @Anton Petrunin.**


Thanks for the help.



length comparison on negatively curved surfaces
This is false even if $K\_{g\_1} = K\_{g\_2} \equiv -1,$ in case the two surfaces are not isometric.


