Smooth Manifolds by Claudio Gorodski

Author:Claudio Gorodski
Language: eng
Format: epub
ISBN: 9783030497750
Publisher: Springer International Publishing

Lemma 3.1.2

Every left invariant vector field X in G is smooth.

Proof

Let f be a smooth function defined on a neighborhood of g in G, and let γ : (−𝜖, 𝜖) → G be a smooth curve with γ(0) = 1 and γ′(0) = X 1. Then the value of X on f is given by

and hence, it is a smooth function of g. □

Let denote the set of left-invariant vector fields on G. It follows that is a vector subspace of . Further, is a subalgebra of for, given X, , we have by Proposition 1.​6.​20 that

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