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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifoldwith nonpostive curvature, then we shall give improved estimates for the$L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics,compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. Byearlier results of Bourgain \cite{bourgainef} and the first author\cite{Sokakeya}, they are equivalent to improvements of the general$L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses thefact that the exponential map from any point in $x_0\in M$ is a universalcovering map from $\Rt \simeq T_{x_0}M$ to $M$ (the Cartan-Hadamard- vonMangolt theorem), which allows us to lift the necessary calculations up to theuniversal cover $(\Rt, \tilde g)$ where $\tilde g$ is the pullback of $g$ viathe exponential map. We then prove the main estimates by using the Hadamardparametrix for the wave equation on $(\Rt, \tilde g)$ and the fact that theclassical comparison theorem of G\"unther \cite{Gu} for the volume element inspaces of nonpositive curvature gives us desirable bounds for the principalcoefficient of the Hadamard parametrix, allowing us to prove our main result.

On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature