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Discrete groups and the geometry of frame bundles. (Diskrete Gruppen und die Geometrie der Repèrebündel.) (German) Zbl 0883.53040

The article studies discrete groups ${\Lambda }$ of isometries of Riemannian manifolds $\left({M}^{n},g\right)$ by lifting their action to the bundle of orthonormal frames $P$. The lifted action leaves a canonical framing invariant. This framing generalizes the left-invariant Maurer-Cartan form on the group of Euclidean motions. Comparison methods are used to extend ideas from the classical theory of crystallographic groups to manifolds with variable curvature.

The main result is the following extension of Gromov’s theorem on almost flat manifolds [M. Gromov, J. Differ. Geom. 13, 231-241 (1978; Zbl 0432.53020)]. Assume that $M$ is simply connected and the quotient space ${\Lambda }\setminus M$ is compact. If the diameter of the quotient and the curvature tensor $R$ of $M$ satisfy the condition ${|R|}_{\infty }\text{diam}\left({\Lambda }\setminus M\right)\le \epsilon \left(n\right)$, then $M$ is diffeomorphic to ${ℝ}^{n}$ and the action of ${\Lambda }$ is isometric for a Riemannian metric ${g}_{0}$ that is left-invariant with respect to a suitable nilpotent Lie group structure on $M$. It follows that a subgroup of index $\le \text{const}\left(n\right)$ is torsion-free. If ${\Lambda }$ is assumed to act freely, this result reduces to Gromov’s theorem in the sharpened form obtained by E. A. Ruh [ibid. 17, 1-14 (1982; Zbl 0468.53036)].

The paper gives a version of the main result in which connections with torsion can replace the Levi-Cività connection of $\left(M,g\right)$. Also, a description of the action of discrete groups near almost fixed points in terms of standard actions on infranilmanifolds is obtained.

Proofs are based on a detailed study of framed Riemannian manifolds, i.e., Riemannian manifolds equipped with an orthonormal parallelization, extending [P. Ghanaat, M. Min-Oo and E. A. Ruh, Indiana Univ. Math. J. 39, 1305-1312 (1990; Zbl 0701.53032)]. This is then applied to the quotient ${\Lambda }\setminus P$, equipped with its canonical framing. The main result is obtained by constructing a nilpotent Maurer-Cartan form on $M$, for which ${\Lambda }$ acts by affine isometries.

##### MSC:
 53C20 Global Riemannian geometry, including pinching