I'm confused what exactly is happening in the following proof of Lemma 4.3 (Lemma 1.5 and Exercise 4.5 are also attached as they are apparently used in this proof). In particular, what does the integration process actually prove? How does it prove what we want? I would imagine that we would take a general geodesic and show that it looks like an exponential.
Is there perhaps a more direct way to prove the result that geodesics in compact Lie groups look like exponential maps? I do apologize in advance - I have little background knowledge in this area of math, and am trying to understand this result as it is used in the proof of Bott-Periodicity. These images are taken from Dietmar Salamon's 'Notes on Compact Lie Groups' (link: https://people.math.ethz.ch/~salamon/PREPRINTS/liegroup.pdf)
They're overcomplicating it. Here's another argument that combines three simple facts.
1) the flows of left-invariant fields consists of right translations; the flows of right-invariant fields consist of left translations.
2) for a bi-invariant metric, both left-invariant fields and right-invariant fields are Killing fields (this follows from item (1)).
3) for any pseudo-Riemannian manifold, the integral curves of constant length Killing fields are geodesics (Killing's equation with $\langle X,X\rangle$ constant implies that $\nabla_XX = 0$).
This means that given $g \in G$ and $v \in T_gG$, we have that $g^{-1}v \doteq {\rm d}(L_{g^{-1}})_g(v) \in \mathfrak{g}$. Then $\alpha(t) = \exp(t g^{-1}v)g$ is the geodesic with $\alpha(0) = g$ and $\alpha'(0) = v$.