Proposition 59

Tag 0AC7, permanent. Cite it as [Bourbaki, Tag 0AC7].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let $G$ be a Lie group, $M$ a manifold of class $C^r$ , $f$ and $g$ mappings of class $C^r$ of $M$ into $G$ and $p$ the canonical surjection of $TM$ onto $M$ .

(i) $(fg)^{-1}.d(fg) = (\mathrm{Ad} \circ g \circ p)^{-1} \circ (f^{-1}.df) + g^{-1}.dg.$

(ii) Writing $h(m) = f(m)^{-1}$ for all $m \in M$ ,

h^{-1}.dh = -(\mathrm{Ad} \circ f \circ p) \circ (f^{-1}.df).

Assertion (i) follows from § 2, no. 2, Proposition 7. Assertion (ii) follows from (i) by putting $g = h$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary 1, § 3
Corollary 2, § 3

Cites

§ 2, no. 2, Proposition 7