A Lie Subgroup is defined as follows:(adsbygoogle = window.adsbygoogle || []).push({});

A Lie subgroup of a Lie group G is (i) an abstract subgroup H that is (ii) an immersed submanifold via the inclusion map such that (iii) the group operations on H are smooth.

While a Lie Group is defined as a group with smooth multiplication and inversion maps, that is also a manifold. One would think that a Lie subgroup would be defined so that it is also a Lie group in it's own right. However, immersed submanifolds are not embedded submanfolds, and thus not in general manifolds in their own right.

What is the reason for this definition?

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# Why are Lie Subgroups Defined as Immersed Submanifolds?

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