# A question on Lie sub-group

Well, definition of Lie subgroup what I know is, a Lie subgroup of a Lie group $G$ is an abstract subgroup $H$ which is an immersed submanifold via the inclusion map so that the group operations on $H$ are $C^{\infty}$.

Could any one make me understand the following with an example?

"Because a Lie subgroup is an immersed submanifold, it need not have the relative topology. In particular, the inclusion map $i:H\rightarrow G$ need not be continuous."

• That statement is not correct. What they probably mean is that the map $H\to G$ may not be a homeomorphism onto its image. – Eric O. Korman Aug 16 '12 at 0:15
• @Eric its a copied statement from the book " Introduction to manifolds" by Loring W Tu, page-152. – Marso Aug 16 '12 at 0:17
• Steve's example illustrates the point of that remark, whatever words are used. And, sometimes, textbooks are "wrong" in some logical sense that is not terribly relevant. Again, the essence of the potential problem already occurs in simple cases, as in the irrational winding on torus in Steve's example. – paul garrett Aug 16 '12 at 0:35

## 1 Answer

I can't help you understand that statement, because I don't understand it: an immersion is a differentiable map with injective derivative, and is in particular continuous. A standard example of a Lie subgroup such that the inverse of the inclusion is not continuous is the image of a line with irrational slope in the two-dimensional compact torus (quotient of the plane by $\mathbb{Z}^2$).

As Paul Garrett mentions in a comment above, this kind of example is almost certainly what Tu had in mind when he wrote the statement you quote. One reason for making the definition this way instead of with a more restrictive definition of Lie subgroup is to be flexible enough to allow Lie subgroups of a Lie group to correspond to Lie subalgebras of its Lie algebra.

• FWIW: The statement in the question doesn't appear to be a faithful quote. In my edition of that book the correct sentence Because a Lie subgroup is an immersed submanifold, it need not have the relative topology. indeed appears on page 167 in section 15.2. However, the bizarre In particular, the inclusion ... need not be continuous. does not seem to be there (it'd be a bit surprising, right?). Also, there is a discussion with pictures of the irrational winding on the torus (what else?) on the very same page. – t.b. Aug 16 '12 at 6:50
• then I need to take a look in the new edition. – Marso Aug 16 '12 at 15:04