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 ${T}_{x}M$ (x$\in M)$ maps ${T}_{x}M$ 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 ${\parallel d\omega \parallel}_{\infty}\xb7diam\left(M\right)<\u03f5\left(n\right)$, then M is diffeomorphic to a nilmanifold. Here $\u03f5$ (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 ${\parallel \xb7\parallel}_{\infty}$ 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.

##### MSC:

22E15 | General properties and structure of real Lie groups |

53B21 | Methods of Riemannian geometry |

22E40 | Discrete subgroups of Lie groups |