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Zbl 0672.22008
Ghanaat, Patrick
Almost Lie groups of type ${\bbfR}\sp n$.
(English)
[J] J. Reine Angew. Math. 401, 60-81 (1989). ISSN 0075-4102; ISSN 1435-5345/e

Let M be a smooth manifold. A parallelization of M is a smooth ${\bbfR}\sp n$-valued 1-form $\omega$ : TM$\to {\bbfR}\sp n$ whose restriction to each tangent space $T\sb xM$ (x$\in M)$ maps $T\sb xM$ isomorphically onto ${\bbfR}\sp n$. We study parallelizations with small exterior derivative. \par The main result is as follows: Let M be a compact connected $C\sp{\infty}$ manifold. If $\omega$ : TM$\to R\sp n$ is a parallelization such that $\Vert d\omega \Vert\sb{\infty}\cdot diam(M)<\epsilon (n)$, 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 $\Vert \cdot \Vert\sb{\infty}$ 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.
[P.Ghanaat]
MSC 2000:
*22E15 General properties and structure of real Lie groups
53B21 Methods of Riemannian geometry
22E40 Discrete subgroups of Lie groups

Keywords: 1-form; parallelizations; small exterior derivative; compact connected $C\sp{\infty }$ manifold; Riemannian metric; nilmanifold; simply connected nilpotent Lie group; discrete uniform subgroup; unimodular Maurer-Cartan equations

Cited in: Zbl 0788.53024

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