  In this paper we deduce a local deformation lemma for uniform embeddings in a
metric covering space over a compact manifold from the deformation lemma for
embeddings of a compact subspace in a manifold. This implies the local
contractibility of the group of uniform homeomorphisms of such a metric
covering space under the uniform topology. Further more, combining with
similarity transformation, this enables us to induce a global homotopy property
of groups of uniform homeomorphisms of metric spaces with Euclidean ends. In
particular, we show that the identity component of the group of uniform
homeomorphisms of the standard Euclidean n-space is contractible.
