Almost flat manifold

Almost flat manifold   (Redirected from Gromov–Ruh theorem) Jump to navigation Jump to search In mathematics, a smooth compact manifold M is called almost flat if for any {displaystyle varepsilon >0} there is a Riemannian metric {displaystyle g_{varepsilon }} on M such that {displaystyle {mbox{diam}}(M,g_{varepsilon })leq 1} and {displaystyle g_{varepsilon }} is {displaystyle varepsilon } -flat, i.e. for the sectional curvature of {displaystyle K_{g_{varepsilon }}} we have {displaystyle |K_{g_{epsilon }}|0} such that if an n-dimensional manifold admits an {displaystyle varepsilon _{n}} -flat metric with diameter {displaystyle leq 1} then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions.

According to the Gromov–Ruh theorem, M is almost flat if and only if it is infranil. In particular, it is a finite factor of a nilmanifold, which is the total space of a principal torus bundle over a principal torus bundle over a torus.

Notes References Hermann Karcher. Report on M. Gromov's almost flat manifolds. Séminaire Bourbaki (1978/79), Exp. No. 526, pp. 21–35, Lecture Notes in Math., 770, Springer, Berlin, 1980. Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp. Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981. Gromov, M. (1978), "Almost flat manifolds", Journal of Differential Geometry, 13 (2): 231–241, doi:10.4310/jdg/1214434488, MR 0540942. Ruh, Ernst A. (1982), "Almost flat manifolds", Journal of Differential Geometry, 17 (1): 1–14, doi:10.4310/jdg/1214436698, MR 0658470.

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