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

Let M be a smooth manifold. A parallelization of M is a smooth ${ℝ}^{n}$-valued 1-form $\omega$ : TM$\to {ℝ}^{n}$ whose restriction to each tangent space ${T}_{x}M$ (x$\in M\right)$ maps ${T}_{x}M$ isomorphically onto ${ℝ}^{n}$. 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 ${\parallel d\omega \parallel }_{\infty }·diam\left(M\right)<ϵ\left(n\right)$, then M is diffeomorphic to a nilmanifold. Here $ϵ$ (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 ${\parallel ·\parallel }_{\infty }$ 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 Riemannian geometry 22E40 Discrete subgroups of Lie groups