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25. Recall that the general theory for a system of ordinary differential equations x′ (t) = f (x(t)), x(0) = a with V ⊂ Rn open, f : V → Rn a C ∞ -map, a ∈ V guarantees that for some ε > 0 this system has a C ∞ -solution t → x(t): (−ε, ε) → V , and that any two solutions defined on the same connected interval containing 0 are equal. If f above moreover has C ∞ -dependence on y ∈ W with W ⊂ Rm open (so (x, y) → f (x, y): V × W → Rn is C ∞ ) then the solution x(t) = x(t, y) also depends on y ∈ W and (t, y) → x(t, y) is C ∞ .

1). 5 Conversely, let be given A ∈ Te G. 1) for t in some interval (−ε, ε). Now we will show that α(s + t) = α(s)α(t) if |s|, |t|, and |s + t| are < ε. Let |s| < ε and put β(t) := α(s + t), γ(t) := α(s) α(t). Then β(0) = α(s), γ(0) = γ(s) and, on the one hand, β ′ (t) = α′ (s + t) = (dℓα(s+t) )e A = (dℓβ(t) )e A, on the other hand, γ ′ (t) = (dℓα(s))α(t) α′ (t) =(dℓα(s) )α(t) (dℓα(t) )e A = d(ℓα(s) ◦ ℓα(t) )e A = (dℓα(s)α(t) )e A = (dℓγ(t) )e A. e’s, so they must be equal. 1) on (−ε, ε) which satisfies α(s+t) = α(s)α(t) = α(t)α(s) for s, t, s + t in the definition interval.

Then the inverse map exp−1 : V → U is denoted by log. Proof The second statement follows by the inverse function theorem. For the proof of the first statement let A ∈ g, α(t) := tA and β(t) := exp(tA). Thus α represents A ∈ T0 g and A = β ′ (0) = d exp0 (A). Hence d exp0 = id. 18 Proposition Let G be a Lie group and put g := Te G. 3) as |A|, |B| → 0 in g. In particular, if G ⊂ GL(n, C ) is a linear Lie group (and thus g ⊂ gl(n, C )) then b(A, B) = AB − BA (A, B ∈ g). Proof Clearly, by the uniqueness of Taylor expansion, the bilinear map b is unique if it exists.