Interestingly, Cheng and Yau were able to use their differential Harnack estimates to show that, under certain geometric conditions, closed submanifolds of complete Riemannian or pseudo-Riemannian spaces are themselves complete. For instance, they showed that if is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then the induced Riemannian metric on is complete. Analogously, they showed that if is an affine hypersphere of affine space which is topologically closed, then the induced affine metric on is complete. Such results are achieved by deriving a differential Harnack inequality for the (squared) distance function to a given point and integrating along intrinsically defined paths.