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I don't understand why the following statement is true.

'A noncompact Lie group need not have any nontrivial tori (e.g. $\mathbb{R}^n$).

Taking $n=2$, we get $\mathbb{R}^2$. Now consider the set of all elements of modulus 1, that is, the unit circle lying in $\mathbb{R}^2$ which is a subgroup isomorphic to $U(1)$. Doesn't this correpond to a non-trivial tori?

Aran
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1 Answers1

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They refer to the additive group $(\Bbb R^n,+)$. The set you describe is not a subgroup; it is not closed under addition.

By the Heine-Borel theorem, the only compact subgroup of $(\Bbb R^n,+)$ is $\{0\}$. Hence, $(\Bbb R^n,+)$ has no non-trivial tori.

Spenser
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