Collettore quasi piatto

Collettore quasi piatto (Redirected from Gromov–Ruh theorem) Vai alla navigazione Vai alla ricerca In matematica, a smooth compact manifold M is called almost flat if for any {displaystyle varepsilon >0} there is a Riemannian metric {stile di visualizzazione g_{varepsilon }} on M such that {stile di visualizzazione {mbox{diam}}(M,g_{varepsilon })leq 1} e {stile di visualizzazione g_{varepsilon }} è {displaystyle varepsilon } -flat, cioè. for the sectional curvature of {stile di visualizzazione K_{g_{varepsilon }}} noi abbiamo {stile di visualizzazione |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. D'altro canto, 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 particolare, 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. Seminario Bourbaki (1978/79), esp. No. 526, pp. 21–35, Lecture Notes in Math., 770, Springer, Berlino, 1980. Peter Buser and Hermann Karcher. Gromov's almost flat manifolds. Asterisco, 81. Société Mathématique de France, Parigi, 1981. 148 pp. Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlino, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981. Gromov, M. (1978), "Almost flat manifolds", Giornale di geometria differenziale, 13 (2): 231–241, doi:10.4310/jdg/1214434488, SIG 0540942. Ruh, Ernst A. (1982), "Almost flat manifolds", Giornale di geometria differenziale, 17 (1): 1–14, doi:10.4310/jdg/1214436698, SIG 0658470.

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