Abstract
In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature ≥1 that are not limits of manifolds with curvature ≥δ > 1/4.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aloff, S., and Wallach, N. L. An infinite family of 7-manifolds admitting positively curved Riemannian structures.BAMS 81, 93–97 (1975).
Browder, W. Higher torsion in H-spaces.TAMS 108, 353–375 (1963).
Burago, Yu., Gromov, M., and Perelman, G., A. D. Alexandrov’s spaces with curvatures bounded from below. To appear inUspekhi Mat. Nauk.
Cheeger, J., and Gromov, M. Collapsing Riemannian manifolds while keeping their curvatures bounded I.J. Diff. Geom. 23, 309–346 (1986).
Freedman, M. The topology of four-manifolds.J. Diff. Geom. 17, 357–453 (1982).
Fukaya, K. Hausdorff convergence of riemannian manifolds and its applications.Adv. Studies in Pure Math. 18-I, 143–238(1990).
Fukaya, K. A boundary of the set of Riemannian manifolds with bounded curvatures and diameters.J. Diff. Geom. 28, 1–21 (1988).
Gromov, M. Groups of polynomial growth and expanding maps.Publ. Math. IHES 53, 53–73 (1981).
Grove, K., and Petersen, P. Manifolds near the boundary of existence.J. Diff. Geom. 33, 379–394 (1991).
Grove, K., and Petersen, P. On the excess of metric spaces and manifolds. Preprint (see also A pinching theorem for homotopy spheres.JAMS 3(3), 671–677 (1990).
Grove, K., Petersen, P., and Wu, J. Y. Geometric finiteness theorems via geometric topology.Invt. Math. 99, 205–213 (1990).Erratum Invt. Math. 104, 221–222 (1991).
Grove, K., and Shiohama, K. A generalized sphere theorem.Ann. Math. 106, 201–211 (1977).
Neill, B. The fundamental equations of submersions.Mich. J. Math. 13, 459–469 (1966).
Perelman, G. Alexandrov’s spaces with curvatures bounded from below II. Preprint.
Petersen, P. Gromov-Hausdorff convergence of metric spaces. To appear inProc. AMS Summer Inst. in Diff. Geom. at UCLA.
Petersen, P., and Zhu, S.-h. An excess sphere theorem. To appear inAnn. Sci. Ec. Norm. Sup.
Smale, S. Generalized Poincare conjecture in dimensions greater than four.Ann. Math. 74, 391–406 (1961).
Thurston, W. Geometry and topology of 3-manifolds. Preprint.
Toponogov, V. Riemannian spaces with curvature bounded below.Uspehi Mat. Nauk 14 (1959) (in Russian).
Yamaguchi, T. Collapsing and pinching under a lower curvature bound.Ann. Math. 133, 317–357 (1991).
Author information
Authors and Affiliations
Additional information
Communicated by Robert Greene
Peter Petersen was supported in part by the National Science Foundation and Alfred P. Sloan Foundation, Frederick Wilhelm was supported by an Alfred P. Sloan doctoral dissertation fellowship, and Shun-hui Zhu was supported in part by the National Science Foundation.
Rights and permissions
About this article
Cite this article
Petersen, P., Wilhelm, F. & Zhu, Sh. Spaces on and beyond the boundary of existence. J Geom Anal 5, 419–426 (1995). https://doi.org/10.1007/BF02921805
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02921805