  A local deformation property for uniform embeddings in metric manifolds (LD)
is formulated and its behaviour is studied in a formal view point. It is shown
that any metric manifold with a geometric group action, typical metric spaces
(Euclidean space, hyperbolic space and cylinders) and for \kappa \leq 0 the
\kappa-cone ends over any compact Lipschitz metric manifolds, all of them have
the property (LD). We also formulate a notion of end deformation property for
uniform embeddings over proper product ends (ED). For example, the 0-cone end
over a compact metric manifold has the property (ED) if it has the property
(LD). It is shown that if a metric manifold M has finitely many proper product
ends with the property (ED), then the group of bounded uniform homeomorphisms
of M endowed with the uniform topology admits a strong deformation retraction
onto the subgroup of bounded uniform homeomorphisms which are identity over
those ends. We also study a role of uniform isotopies in deformation of uniform
homeomorphisms and show that Alexander isotopies in \kappa-cones induce
contractions of some subgroups of groups of bounded uniform homeomorphisms.
