This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all eigenvalues of the second fundamental form, with respect to a particular unit normal field, be at least one. The following alternative must hold: either there is a point where all the eigenvalues of the second fundamental form are strictly greater than one, in which case the hypersurface is

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... imbedded and diffeomorphic to a sphere; or, the second fundamental form at every point has 1 as an eigenvalue, in which case the hypersurface is isometric to Euclidean space and is imbedded in hyperbolic space as a horosphere. This paper is devoted to the consideration of complete, immersed, C°°h ypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. By the latter phrase, we mean that, with respect to the appropriately chosen normal, the second fundamental form is greater than or equal to the identity, the second fundamental form of a horosphere. Equivalently, all the eigenvalues of the second fundamental form of the hypersurface are greater than or equal to 1. The results are given in the following theorems: Theorem A. Suppose M is a complete, connected C°° Riemannian manifold whose dimension, n, is at least 2. Suppose that f: M -* Hn+ is a C°°i sometric immersion and that there exists a smooth normal field v along f such that, at every point of M, all the eigenvalues of the second fundamental form of M in H"+ ' with respect to v are at least 1. If there is a point p in M where all the eigenvalues of II are strictly greater than 1, then ( 1 ) M is compact, (2) M is imbedded as the boundary of a convex body, and (3) M is diffeomorphic to S". Theorem B. Suppose that M, f and v are as in Theorem A. If the least eigenvalue of II at every point is 1, then (1) M is isometric to R" , and (2) M is imbeded as a horosphere. The Gauss equation implies that such a hypersurface has nonnegative intrinsic sectional curvature. Thus, Theorems A and B form a partial analogue of