Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I would like to review a classical result by Milnor, Curvatures of left invariant metrics on Lie groups. J., Adv. Math. 21 (1976), no. 3, 293-329.

Theorem 1.5 (page 298). A Lie group with left invariant metric is flat if and only if the associated Lie algebra splits as an orthogonal direct sum $\mathfrak{b}\oplus\mathfrak{u}$ where $\mathfrak{b}$ is a commutative sub algebra, $\mathfrak{u}$ is a commutative ideal, and where the linear transformation $\mathrm{ad}(b)$ is skew-adjoint for every $b\in\mathfrak{b}$.

In the proof (page 326), Milnor uses two geometrical results: Cauchy-Hadamard theorem and Myers theorem.

My question is as follows:

Is there any algebraic proof of Milnor's theorem? May one uses Chevalley-Eilenberg cohomology or such algebraic tools to give a 'new' proof to the result aboe?

Although, I proved (with Pr. Boucetta) an improved version of Milnor's theorem. See proposition 2.1 and 2.2, page 5.

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.