## Almost Lie groups of type $${\mathbb{R}}^ n$$.(English)Zbl 0672.22008

Let M be a smooth manifold. A parallelization of M is a smooth $${\mathbb{R}}^ n$$-valued 1-form $$\omega$$ : TM$$\to {\mathbb{R}}^ n$$ whose restriction to each tangent space $$T_ xM$$ (x$$\in M)$$ maps $$T_ xM$$ isomorphically onto $${\mathbb{R}}^ n$$. We study parallelizations with small exterior derivative.
The main result is as follows: Let M be a compact connected $$C^{\infty}$$ manifold. If $$\omega$$ : TM$$\to R^ n$$ is a parallelization such that $$\| d\omega \|_{\infty}\cdot diam(M)<\epsilon (n)$$, then M is diffeomorphic to a nilmanifold. Here $$\epsilon$$ (n) is a positive constant depending only on the dimension n of M, diam(M) is the diameter of M with respect to the Riemannian metric induced by $$\omega$$ and $$\| \cdot \|_{\infty}$$ is the maximum norm on tensors with respect to that metric. We recall that a nilmanifold is a quotient of a simply connected nilpotent Lie group by a discrete uniform subgroup. The proof uses an iterated variational method to deform the given $$\omega$$ into a solution of the unimodular Maurer-Cartan equations.
Reviewer: P.Ghanaat

### MSC:

 22E15 General properties and structure of real Lie groups 53B21 Methods of local Riemannian geometry 22E40 Discrete subgroups of Lie groups
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