A half-space property for hypersurfaces in the hyperbolic space

Through the study of geometry of the hyperspheres (also known as equidistant spheres) of the hyperbolic space $\mathbb H^{n+1}$, we establish a nonexistence result for complete noncompact hypersurfaces immersed into $\mathbb H^{n+1}$ and a characterization of complete totally geodesic hypersurfaces of $\mathbb H^{n +1}$; namely, we characterize those complete hyperspheres of $\mathbb H^{n +1}$ with the following geometric property: any geodesic contained in a complete hypersurface is also a geodesic of $\mathbb H^{n+1}$. Our approach is based on a suitable maximum principle at infinity for complete Riemannian manifolds.