We first recall a theorem proved in “Exponential maps on Lie groups”:

**Ken’s Theorem**: Let G be a Lie groups with a bi-invariant metric (e.g. if G is compact). Then the exponential map is the same as the Riemannian exponential map

* at the identity .*

In this note **we assume that G is compact**, and we wish to derive some consequences of the above theorem. The most important one is the

**Cartan’s theorem**proved below. Some information concerning the fundamental group of G can also be obtained. One might consult the two previous blogs by KKK on Lie groups for the background information needed.

First we state a lemma.

* Lemma 1: The exponential map is surjective*.

The lemma is a direct consequence of Ken’s theorem, because any metric defined on a compact manifold is complete, and completeness is equivalent to the fact that

is surjective.

Next we introduce the notion of a maximal torus. By definition, a maximal torus *T* in *G* is a maximal connected abelian subgroup of *G*. Here maximal means that if *T’* contains *T* and *T’* is connected and abelian, then . One can check that *T* must be closed (because the closure of any subgroup is still a subgroup), and *T* must be a torus

by compactness of *G* (the proof is omitted, it depends on some analysis on the exponential map).

Note that maximal tours are not unique: if *T* is a maximal torus, then is also a maximal torus. Cartan’s theorem states that all maximal torus are of this form:

**Cartan’s theorem**: Let T, T’ be two maximal torus. Then there is a such that *.*

We first state a proposition.

**Proposition 1**: Let T be a maximal torus and . Then such that .

*Proof of proposition 1*: Let and be a generator of *T*. Here a generator means that (why exists?). Using Lemma 1, there is and such that and . Let be choosen such that is maximized.

Let (Then is a generator of ). For any , the term has a maximum at . Thus

.

The proof for the second and thrid equalities can be found in KKK’s “Lie groups with bi-invariant Riemannian metric”. As *Y* is arbitrary, and . Thus for any *n* and so commutes with every elements in . Let *W* be the connected abelian group generated by and . As is maximal abelian, we have and .

*Proof of Cartan’s theorem*: Let *t’* be a generator of *T’*. Then for some *x, *by proposition 1. Then . As *T’* is maximal abelian, and the theorem is proved.

Indeed, Cartan’s theorem is quite important in representation theory of compact Lie group. As *T* is abelian, representation on *T* is well known (just Fourier analysis). One hopes to know all representations of *G* by looking at that of* T*, and Cartan’s theorem tells us that all maximal abelian connected subgroups *T* are the same.

To end this note, we give an application of Cartan’s theorem. Let *T* be a maximal torus and

be the inclusion map.

**Theorem 2**: The induced map is surjective. Thus the fundamental group of G is abelian and finitely generated.

*Proof of theorem 2*: Let . Then by compactness of G, can be choosen to be a geodesic (see Jost for a detailed discription, at least it is intuitively clear…). By a left multiplication (which is an isometry as the metric is bi-invariant) if neccesary, we can assumed that this geodesic starts at the identity e in G. Then for some . Note that this subset forms an abelian subgroup of G, thus is contained in some maximal torus T’. By Cartan’s theorem, for some x and

,

where . Let be a path joining *e* to* x*. Then obviously

forms a homotopy between and . Thus and the theorem is proved.

To avoid confusion, in the proof of Proposition 1, should be written as instead (there are two instances).

Changed as suggested, Thank you~