  In this paper we consider a family of Riemannian manifolds, not necessarily
complete, with curvature conditions in a neighborhood of a ray. Under these
conditions we obtain that the essential spectrum of the Laplacian contains an
interval. The results presented in this paper allow to determine the spectrum
of the Laplace operator on unlimited regions of space forms, such as horoball
in hyperbolic space and cones in Euclidean space.
