By L. Pontrjagin

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29. Examples. (1) The singular distribution spanned by W ⊂ Xloc (R2 ) is involutive, but not integrable, where W consists of all global vector fields with ∂ support in R2 \ {0} and the field ∂x 1 ; the leaf through 0 should have dimension 1 at 0 and dimension 2 elsewhere. ∂ (2) The singular distribution on R2 spanned by the vector fields X(x1 , x2 ) = ∂x 1 ∂ and Y (x1 , x2 ) = f (x1 ) ∂x2 where f : R → R is a smooth function with f (x1 ) = 0 for x1 ≤ 0 and f (x1 ) > 0 for x1 > 0, is involutive, but not integrable.

Our first aim is to show that ϕ is a homeomorphism. K −1 ⊂ V . K. , Vi open and dense for i ∈ N implies Vi dense). The set ϕ(ai )ϕ(K) is compact, thus closed. ϕ(K), there is some i such that ϕ(ai )ϕ(K) has non empty interior, so ϕ(K) has non empty interior. Choose b ∈ G such that ϕ(b) is an interior point of ϕ(K) in H. Then eH = ϕ(b)ϕ(b−1 ) is an interior point of ϕ(K)ϕ(K −1 ) ⊂ ϕ(V ). So if U is open in G and a ∈ U , then eH is an interior point of ϕ(a−1 U ), so ϕ(a) is in the interior of ϕ(U ).

If X, Y , and t are small enough we get ad C(t) = log(et. C(t) = g(log(et. C(t). For z near 1 we put f (z) := 1. f (z) = ˙ ˙ X = g(log(et. C(t) = f (et. C(t), ˙ C(t) = f (et. X, C(0) = Y Passing to the definite integral we get the desired formula 1 C(X, Y ) = C(1) = C(0) + ˙ dt C(t) 0 1 f (et. ℓn ≥0 ki +ℓi ≥1 tk (ad X)k (ad Y )ℓ k! ℓ! n X dt (ad X)k1 (ad Y )ℓ1 . . (ad X)kn (ad Y )ℓn X (k1 + · · · + kn + 1)k1 ! . ℓ1 ! . ℓn ! 1 12 ([X, [X, Y ]] − [Y, [Y, X]]) + · · · Remark. If G is a Lie group of differentiability class C 2 , then we may define T G and the Lie bracket of vector fields.

### Topologische Gruppen, Teil 2 by L. Pontrjagin

by Kenneth

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