By Bryant R.L.

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Once this is done, the rest of the construction of charts with smooth overlaps follows the end of the proof of Theorem 1 almost verbatim. 6 43 Group Actions and Vector Fields. A left action λ: R × M → M (where R has its usual additive Lie group structure) is, of course, the same thing as a ﬂow. Associated to each ﬂow on M is a vector ﬁeld which generates this ﬂow. The generalization of this association to more general Lie group actions is the subject of this section. Let λ: G × M → M be a left action.

Finally, since the map v → Yv is [Yx , Yy ] = −Y[x,y] and hence must be equal to −Y[x,y] clearly linear, it follows that λ∗ is also linear. The appearance of the minus sign in the above formula is something of an annoyance and has led some authors (cf. [A]) to introduce a non-classical minus sign into either the deﬁnition of the Lie bracket of vector ﬁelds or the deﬁnition of the Lie bracket on g in order to get rid of the minus sign in this theorem. Unfortunately, as logical as this revisionism is, it has not been particularly popular.

Let λ: G × M → M be a left action. Then, for each v ∈ g, there is a ﬂow Ψλv on M deﬁned by the formula Ψλv (t, m) = etv · m. This ﬂow is associated to a vector ﬁeld on M which we shall denote by Yvλ , or simply Yv if the action λ is clear from context. This deﬁnes a mapping λ∗ : g → X(M), where λ∗ (v) = Yvλ . Proposition 1: For each left action λ: G × M → M, the mapping λ∗ is a linear antihomomorphism from g to X(M). In other words, λ∗ is linear and λ∗ ([x, y]) = −[λ∗ (x), λ∗ (y)]. Proof: For each v ∈ g, let Yv denote the right invariant vector ﬁeld on G whose value at e is v.