The global asymptotic stability conjecture in dynamical systems was solved recently and independently by Feller, Glutsiuk and Gutierrez. Crucial to the approach of Gutierrez is the following theorem of his: A local diffeomorphism $f:\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ for which the eigenvalues of $Df(x)$ miss $(0,\infty)$ must be injective. The present paper gives a partial generalization of this theorem to local diffeomorphisms between Hadamard surfaces, the spectral condition being replaced by transversality conditions among certain foliations associated to horocycles. The proofs use arguments from global analysis.