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Almost Lie groups of type n . (English) Zbl 0672.22008

Let M be a smooth manifold. A parallelization of M is a smooth n -valued 1-form ω : TM n whose restriction to each tangent space T x M (xM) maps T x M isomorphically onto n . We study parallelizations with small exterior derivative.

The main result is as follows: Let M be a compact connected C manifold. If ω : TMR n is a parallelization such that dω ·diam(M)<ϵ(n), 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 ω and · 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 ω into a solution of the unimodular Maurer-Cartan equations.

Reviewer: P.Ghanaat

MSC:
22E15General properties and structure of real Lie groups
53B21Methods of Riemannian geometry
22E40Discrete subgroups of Lie groups