Abstract
Let M n be an n(≧ 3)-dimensional compact, simply connected Riemannian manifold without boundary and S n be the unit sphere of the Euclidean space R n+1. By two different means we derive an estimate of the diameter whenever the manifold considered satisfies that the sectional curvature K M ≦ 1, while Ric (M) ≧ \(\tfrac{{n + 2}}{4}\) and the volume V (M) ≦ \(\tfrac{3}{2}\)(1 + η)V (S n) for some positive number η depending only on n. Consequently, a gap phenomenon of the manifold will be given according to the estimate of the diameter.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland (Amsterdam, 1975).
L. Coghlan and Y. Itokawa, A sphere theorem for reverse volume pinching on even-dimension manifolds, Proc. Amer. Math. Soc., 111 (1991), 815–819.
J.-Y. Wu, Hausdorff convergence and sphere theorem, Proc. Sym. Pure Math., 54 (1993), 685–692.
Yuliang Wen, A note on pinching sphere theorem, C.R. Acad. Sci. Paris, Ser. I, 338 (2004), 229–234.
K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math., 106 (1977), 201–211.
K. Pawel, On the spectral gap for compact manifolds, J. Differential Geom., 36 (1992), 315–330.
T. Sakai, Riemannian Geometry, Shokabo Publishing Co. Ltd. (Tokyo, 1992).
K. Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc., 27 (1983), 811–819.
J. Y. Wu, A diameter pinching sphere theorem, Proc. Amer. Math. Soc., 197 (1990), 796–802.
C. Y. Xia, Rigidity and sphere theorem for manifolds with positive Ricci curvature, Manuscript Math., 85 (1994), 79–87.
K. Grove and V. P. Petersen, Manifolds near the boundary of existence, J. Diff. Geom., 33 (1991), 379–394.
P. Hartman, Oscillation criteria for self-adjoint second-order differential systems and principle sectional curvature, J. Diff. Equations, 33 (1991), 379–394.
S. Peters, Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 3–16.
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Pub. Math., 53 (1981), 183–215.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser (Boston, 1999).
Author information
Authors and Affiliations
Additional information
Supported by the NNSF of P.R. China (No. 10371039) and Scientific Research start-up Foundation of QFNU and Shandong Priority Academic Discipline.
Rights and permissions
About this article
Cite this article
Wang, P. A gap phenomenon on Riemannian manifolds with reverse volume pinching. Acta Math Hung 115, 133–144 (2007). https://doi.org/10.1007/s10474-006-0536-4
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10474-006-0536-4