Weakened Lie groups and their locally isometric completions

Weakened Lie groups are Lie groups with a Hausdorff topology that is weaker than the Lie topology. We show that a large class of weakened Lie groups are locally isometric. If the weakened groups are not complete (and they usually are not), then the same property holds for their completions. This is a surprising result since, on a global scale, the weakened groups may exhibit many “unusual” and distinct characteristics. Other results include a constructive procedure for obtaining metrizable, weakened Lie groups and examples of metrizable topological groups with unusual properties.