Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces. (English) Zbl 1192.53040

A closed smooth manifold \(M\) of dimension \(m\) is said to be almost nonnegatively curved if it can Gromov-Hausdorff converge to a single point under a lower curvature bound, i.e., if it admits a sequence of Riemannian metrics with sectional curvatures bounded from below by \(-1/n\) and diameters bounded from above by \(1/n\). The authors show:
Theorem A. Let \(M\) be a closed almost nonnegatively curved manifold. Then a finite cover of \(M\) is a nilpotent space. The authors also provide an affirmative answer to a conjecture of K. Fukaya and T. Yamaguchi [Ann. Math. (2) 136, No. 2, 253–333 (1992; Zbl 0770.53028)].
Theorem B. Let \(M\) be an almost nonnegatively curved \(m\)-manifold. Then \(\pi_1(M)\) contains a nilpotent subgroup of index at most \(C(m)\). The authors also establish a fibration theorem.
Theorem C. Let \(M\) be an almost nonnegatively curved manifold. Then a finite cover \(\tilde M\) of \(M\) is the total space of a fiber bundle \(F\rightarrow\tilde M\rightarrow N\) over a nilmanifold \(N\) with simply connected fiber \(F\). Moreover, the fiber \(F\) is almost nonnegatively curved in a generalized sense.
The authors use Alexandrov geometry and the gradient flow of the square of a distance function. They use “limit fundamental groups of Alexandrov spaces” and discuss some open questions.


53C20 Global Riemannian geometry, including pinching


Zbl 0770.53028
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[1] M. T. Anderson, ”Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem,” Duke Math. J., vol. 68, iss. 1, pp. 67-82, 1992. · Zbl 0767.53029
[2] Y. Burago, M. Gromov, and G. Perelcprimeman, ”A. D. Aleksandrov spaces with curvatures bounded below,” Uspekhi Mat. Nauk, vol. 47, iss. 2(284), pp. 3-51, 222, 1992.
[3] J. Cheeger, K. Fukaya, and M. Gromov, ”Nilpotent structures and invariant metrics on collapsed manifolds,” J. Amer. Math. Soc., vol. 5, iss. 2, pp. 327-372, 1992. · Zbl 0758.53022
[4] K. Fukaya and T. Yamaguchi, ”The fundamental groups of almost non-negatively curved manifolds,” Ann. of Math., vol. 136, iss. 2, pp. 253-333, 1992. · Zbl 0770.53028
[5] S. Gallot, ”Inégalités isopérimétriques, courbure de Ricci et invariants géométriques, II,” C. R. Acad. Sci. Paris Sér. I Math., vol. 296, iss. 8, pp. 365-368, 1983. · Zbl 0535.53035
[6] K. Grove and S. Halperin, ”Contributions of rational homotopy theory to global problems in geometry,” Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 171-177, 1982. · Zbl 0508.55013
[7] M. Gromov, ”Almost flat manifolds,” J. Differential Geom., vol. 13, iss. 2, pp. 231-241, 1978. · Zbl 0432.53020
[8] M. Gromov, ”Synthetic geometry in Riemannian manifolds,” in Proc. Internat. Congress of Mathematicians, Helsinki, 1980, pp. 415-419. · Zbl 0427.53018
[9] M. Gromov, ”Curvature, diameter and Betti numbers,” Comment. Math. Helv., vol. 56, iss. 2, pp. 179-195, 1981. · Zbl 0467.53021
[10] M. Gromov, ”Volume and bounded cohomology,” Inst. Hautes Études Sci. Publ. Math., iss. 56, pp. 5-99, 1982. · Zbl 0516.53046
[11] K. Grove, ”Geometry of, and via, symmetries,” in Conformal, Riemannian and Lagrangian Geometry, Providence, RI: Amer. Math. Soc., 2002, pp. 31-53. · Zbl 1019.53001
[12] P. Hilton, G. Mislin, and J. Roitberg, Localization of Nilpotent Groups and Spaces, Amsterdam: North-Holland, 1975. · Zbl 0323.55016
[13] C. LeBrun, ”Ricci curvature, minimal volumes, and Seiberg-Witten theory,” Invent. Math., vol. 145, iss. 2, pp. 279-316, 2001. · Zbl 0999.53027
[14] A. Tralle and J. Oprea, Symplectic Manifolds with no Kähler Structure, New York: Springer-Verlag, 1997. · Zbl 0891.53001
[15] G. Perelman, ”Alexandrov spaces with curvatures bounded from below, II,” , preprint , 1991.
[16] A. Petrunin, ”Quasigeodesics in multidimensional Alexandrov spaces,” PhD Thesis , University of Illinois at Urbana Champaign, 1995.
[17] A. Petrunin, ”Parallel transportation for Alexandrov space with curvature bounded below,” Geom. Funct. Anal., vol. 8, iss. 1, pp. 123-148, 1998. · Zbl 0903.53045
[18] A. Petrunin, ”Semiconcave functions in Alexandrov’s geometry,” , to appear in Surveys in Comparison Geometry , 2007. · Zbl 1166.53001
[19] G. Perelman and A. Petrunin, ”Quasigeodesics and gradient curves in Alexandrov spaces,” , preprint , 1996.
[20] G. P. Paternain and J. Petean, ”Zero entropy and bounded topology,” Comment. Math. Helv., vol. 81, iss. 2, pp. 287-304, 2006. · Zbl 1093.53090
[21] X. Rong, ”On the fundamental groups of manifolds of positive sectional curvature,” Ann. of Math., vol. 143, iss. 2, pp. 397-411, 1996. · Zbl 0974.53029
[22] X. Rong, ”The almost cyclicity of the fundamental groups of positively curved manifolds,” Invent. Math., vol. 126, iss. 1, pp. 47-64, 1996. · Zbl 0859.53022
[23] A. Silverberg and Y. G. Zarhin, ”Variations on a theme of Minkowski and Serre,” J. Pure Appl. Algebra, vol. 111, iss. 1-3, pp. 285-302, 1996. · Zbl 0885.14006
[24] R. M. Schoen, ”Variational theory for the total scalar curvature functional for Riemannian metrics and related topics,” in Topics in Calculus of Variations, Giaquinta, M., Ed., New York: Springer-Verlag, 1989, pp. 120-154. · Zbl 0702.49038
[25] V. A. Sharafutdinov, ” The Pogorelov-Klingenberg theorem for manifolds homeomorphic to \(\mathbbR^n\),” Sib. Math. J., vol. 18, pp. 649-657, 1978. · Zbl 0411.53031
[26] L. J. Schwachhöfer and W. Tuschmann, ”Metrics of positive Ricci curvature on quotient spaces,” Math. Ann., vol. 330, iss. 1, pp. 59-91, 2004. · Zbl 1062.53027
[27] D. Sullivan, ”Infinitesimal computations in topology,” Inst. Hautes Études Sci. Publ. Math., iss. 47, pp. 269-331 (1978), 1977. · Zbl 0374.57002
[28] B. Wilking, ”On fundamental groups of manifolds of nonnegative curvature,” Differential Geom. Appl., vol. 13, iss. 2, pp. 129-165, 2000. · Zbl 0993.53018
[29] T. Yamaguchi, ”Collapsing and pinching under a lower curvature bound,” Ann. of Math., vol. 133, iss. 2, pp. 317-357, 1991. · Zbl 0737.53041
[30] S. Ohta, ”Gradient flows on Wasserstein spaces over compact Alexandrov spaces,” Amer. J. Math., vol. 131, iss. 2, pp. 475-516, 2009. · Zbl 1169.53053
[31] G. Savaré, ”Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds,” C. R. Math. Acad. Sci. Paris, vol. 345, iss. 3, pp. 151-154, 2007. · Zbl 1125.53064
[32] A. Lytchak, ”Open map theorem for metric spaces,” Algebra i Analiz, vol. 17, iss. 3, pp. 139-159, 2005. · Zbl 1152.53033
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