On Large Scale Properties of Manifolds

We show that a space with a finite asymptotic dimension is embeddable in a non-positively curved manifold. Then we prove that if a uniformly contractible manifold X is uniformly embeddable in $\R^n$ or non-positively curved n-dimensional simply connected manifold then $X\times\R^n$ is integrally hyperspherical. If a uniformly contractible manifold X of bounded geometry is uniformly embeddable into a Hilbert space, then X is stably integrally hyperspherical.