Hostname: page-component-74656f9884-2dmjx Total loading time: 0 Render date: 2026-11-11T02:19:46.817Z Has data issue: false hasContentIssue false
  • English
  • Français

The C1,1 conclusions in Gromov's theory

Published online by Cambridge University Press:  19 September 2008

Charles C. Pugh
Charles C. Pugh
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

According to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 1 , March 1987 , pp. 133 - 147
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[1]Bishop, R. & Crittenden, R.. Geometry of Manifolds, Academic Press, N.Y., 1964.Google Scholar
[2]Cheeger, J. & Ebin, D.. Comparison Theorems in Riemannian Geometry, Amsterdam, 1975.Google Scholar
[3]Cheeger, J. & Gromov, M.. On the characteristic numbers of complete manifolds of bounded curvature and finite volume, In Differential Geometry and Complex Analysis, ed. Chavel, I. & Farkas, H. M.. Springer-Verlag, Berlin, 1985.Google Scholar
[4]Durumeric, O.. A generalization of Berger's almost ¼-pinched theorem, I, Bull. Amer. Math. Soc. 12 (1985), 260–264.CrossRefGoogle Scholar
[5]Greene, R. & Wu, H.. Function Theory on Manifolds which Possess a Pole. Springer Notes #699, Springer-Verlag, Berlin, 1979.CrossRefGoogle Scholar
[6]Greene, R. & Wu, H.. Lipschitz convergence of Riemannian manifolds. Preprint, 1985.Google Scholar
[7]Gromoll, D., Klingenberg, W., & Meyer, W.. Riemannsche Geometrie im Grössen, Springer Notes #55, Springer-Verlag, Berlin, 1968.Google Scholar
[8]Gromov, M.. Structures metriques pour les variétés Riemannienes. ed. Lafontaine, J. & Pansu, P., Textes Math. no. 1, Cedic-Nathan, Paris, 1981.Google Scholar
[9]Hart, D.. On the smoothness of generators. Topology 22 (1983), pp. 357–363.CrossRefGoogle Scholar
[10]Hartman, P.. Ordinary Differential Equations. Birkhauser, Boston, 1982.Google Scholar
[11]Hirsch, M. & Pugh, C.. Smoothness of horocycle foliations. J. Diff. Geom. 10 (1975), pp. 225–238.Google Scholar
[12]Hirsch, M., Pugh, C., & Shub, M.. Invariant Manifolds. Springer Notes # 583, Springer-Verlag, Berlin, 1977.CrossRefGoogle Scholar
[13]Ishihara, S. & Yano, K.. Tangent and Cotangent bundles. Marcel Dekker, N.Y., 1973.Google Scholar
[14]Jost, J. & Karcher, H.. Geometrische methoden zur gewinnung vor. a-priori schranker fur harmonische abbildungen, Man. Math. 40 (1982), pp. 27–77.CrossRefGoogle Scholar
[15]Katsuda, A.. Gromov's convergence theorem and its application. To appear in Nagoya Math. J.Google Scholar
[16]Peters, S.. Cheeger's finiteness theorem for diffeomorphism classes on Riemannian manifolds. J. Reine Angew. Mat. 349 (1984), pp. 77–82.Google Scholar
[17]Peters, S.. Preprint.Google Scholar
[18]Weatherburn, C.. An Introduction to Riemannian Geometry and the Tensor Calculus. Cambridge University Press, Cambridge, 1938.Google Scholar
[19]Whitney, H.. Geometric Integration Theory. Princeton University Press, Princeton, 1957.Google Scholar