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


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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