If G is an arcwise connected,
finite-dimensional group, then it is known that there exists a connected Lie group G∼
and a continuous, one-one, onto homomorphism i : G∼→ G. Results are obtained for
describing the topology of G in terms of the geometry of the manifold G∼ and of i.
The major result is that there is a closed subgroup S of G∼ and arbitrarily small
neighborhoods U at the identity of G such that there are real numbers r and s
satisfying i−1(U) is the union of mutually disjoint open balls in G∼ of radius r
scattered along the submanifold S in such a manner that the balls are separated
by at least s. In the case that G is embedded in a locally compact group,
more detailed information is given for the distribution of the open balls in
G∼.
