Rough Isometries, in the sense of M. Kanai [16], preserve geometric properties of Riemannian manifolds such as volume growth rate, isoperimetric dimensions and Sobolev constants, Liouville property, transience of Brow nian motion and Harnack inequalities. In this manuscript we first look at Asymptotic Ends. We show that in a Riemannian manifold asymptotic ends are preserved under rough isometries. Secondly, we study Mappings with Maximal Rank . Here we use the jargon of fiber bundles. These projection-types define a decomposition of each tangent space to the domain, into the orthogonal sum of Horizontal plus Vertical subspaces, where vertical and horizontal mean tangent to the fibers and orthogonal to the fibers, respectively. Motivated by [25] we investigated the question: when is a Riemannian manifold roughly isometric to a Riemannian product manifold? Firstly, for Riemannian submersions, a concept defined in [25], we show that, if the base manifold is compact then the fibers can be roughly isometrically immersed into the domain, and thus the domain is roughly isometric to the product of any fiber and the base space. Under assumptions on the fibers the Riemannian submersion is a rough isometry, and if a fixed fiber is compact then the domain is roughly isometric to the product of that fiber and the base space. For onto maximal rank maps that are not necessarily submersions, by adding assumptions on the Horizontal Vectors Space we have the same consequences. We provide Counterexamples to show that if any of the assumptions are removed those results cease to follow. Finally, as an answer to the question above we provide the main result. We show that for maximal rank onto mappings between Riemannian manifolds with bounded geometry, under assumptions on the fibers and assumptions on the subspaces of horizontal vectors, the domain of such mapping is roughly isometric to the product of the base manifold and a fixed fiber of the domain.
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