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Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositivesectional curvature. Let $\Delta_g$ be the Laplace-Beltrami operatorcorresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalizedeigenfunctions of $\Delta_g$ with eigenvalue $\lambda$, i.e. \[ -\Delta_ge_\lambda = \lambda^2 e_\lambda. \] We prove \[ \left| \int_{\mathbb R} b(t)e_\lambda (\gamma(t)) \, dt \right| = o(1) \quad \text{ as } \lambda \to \infty\] where $b$ is a smooth, compactly supported function on $\mathbb R$ and$\gamma$ is a curve parametrized by arc-length whose geodesic curvature$\kappa(\gamma(t))$ avoids two critical curvatures $\mathbfk(\gamma'^\perp(t))$ and $\mathbf k(-\gamma'^{\perp}(t))$ for each $t \in\operatorname{supp} b$. $\mathbf k(v)$ denotes the curvature of a circle withcenter taken to infinity along the geodesic ray in direction $-v$.

Integrals of eigenfunctions over curves in surfaces of nonpositive curvature