Ricci curvature, diameter and optimal volume bound

Jyh-Yang WU

doi:10.2969/jmsj/04730537

Jyh-Yang WU

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JOURNAL FREE ACCESS

1995 Volume 47 Issue 3 Pages 537-550

DOI https://doi.org/10.2969/jmsj/04730537

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Published: 1995 Received: June 12, 1993 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : A1) M. Anderson, Short geodesics and gravitational instantons, J. Differential Geom., 31 (1990), 265-275.
A2) M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102 (1990), 427-445.
A3) M. Anderson, Metrics of positive Ricci curvature with large diameter, preprint.
A4) M. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology, 29 (1990), 41-55.
AC) M. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and L^n/2-norm of curvature bounded, J. Differential Geom., 35 (1992), 265-281.
B) A. Besse, Manifolds all of whose geodesics are closed, Springer-Verlag, 1978.
BC) R. Bishop and R. L. Crittendon, Geometry of manifolds, Academic Press, New York, 1974.
BZ) Yu. Burago and V. Zalgaller, Geometric inequalities, Springer-Verlag, New York,1988.
C) C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13 (1980), 419-435.
CE) J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam and New York, 1975.
CG) J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom., 6 (1971), 119-128.
Ch) S. Y. Cheng, Eigenvalue comparison theorem and its geometric applications, Math. Z., 143 (1975), 289-297.
Dk) D. Deturck and J. Kazdan, Regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup., 14 (1980), 249-260.
Ga) L. Gao, Convergence of Riemannian manifolds, Ricci pinching and L^n/2-curvature pinching, J. Differential Geom., 32 (1990), 349-381.
Gr) M. Gromov, Structures metrique pour les variétés Riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
GP) K. Grove and P. Petersen V, Manifolds near the boundary of existence, J.Differential Geom., 33 (1991), 379-394.
GT) D. Gilbarg and N. S. Trüdinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, 1977.
Mi) J. Milnor, Morse theory, Ann. of Math. Stud., No. 51, Princeton University Press, 1963.
Mo) C. Morry, Multiple integrals in the calculus of variations, Springer-Verlag, Heidelberg, 1966.
O) Y. Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z., 206 (1991), 255-264.
P) S. Peters, Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 3-16.
S) T. Sakai, On the continuity of injectivity radius function, Math. J. Okayama Univ., 25 (1983), 91-97.
Sh) K. Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc., 275 (1983), 811-819.
SS) J. Schouten and D. Struik, On some properties of general manifolds relating to Einstein's theory of gravitation, Amer. J. Math., 43 (1921), 213-216.
We) A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom., 9 (1974), 513-517.
Wu1) J.-Y. Wu, The volume/diameter ratio for positively curved manifolds, Michigan Math. J., 37 (1990), 235-239.
Wu2) J.-Y. Wu, Convergence of Riemannian 3-manifolds under Ricci curvature bound, Amer. J. Math., 116 (1994), 1019-1029.
Yg) C. T. Yang, Odd-dimensional Wiedersehen manifolds are spheres, J. Differential Geom., 15 (1980), 91-96.

Right : [A1] M. Anderson, Short geodesics and gravitational instantons, J. Differential Geom., 31 (1990), 265-275.
[A2] M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102 (1990), 427-445.
[A3] M. Anderson, Metrics of positive Ricci curvature with large diameter, preprint.
[A4] M. Anderson, On the topology of complete manifolds of nonnegative Ricci curvature, Topology, 29 (1990), 41-55.
[AC] M. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and L^n/2-norm of curvature bounded, J. Differential Geom., 35 (1992), 265-281.
[B] A. Besse, Manifolds all of whose geodesics are closed, Springer-Verlag, 1978.
[BC] R. Bishop and R. L. Crittendon, Geometry of manifolds, Academic Press, New York, 1974.
[BZ] Yu. Burago and V. Zalgaller, Geometric inequalities, Springer-Verlag, New York, 1988.
[C] C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup., 13 (1980), 419-435.
[CE] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam and New York, 1975.
[CG] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom., 6 (1971), 119-128.
[Ch] S. Y. Cheng, Eigenvalue comparison theorem and its geometric applications, Math. Z., 143 (1975), 289-297.
[Dk] D. Deturck and J. Kazdan, Regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup., 14 (1980), 249-260.
[Ga] L. Gao, Convergence of Riemannian manifolds, Ricci pinching and L^n/2-curvature pinching, J. Differential Geom., 32 (1990), 349-381.
[Gr] M. Gromov, Structures metrique pour les variétés Riemanniennes, Cedic/Fernand Nathan, Paris, 1981.
[GP] K. Grove and P. Petersen V, Manifolds near the boundary of existence, J. Differential Geom., 33 (1991), 379-394.
[GT] D. Gilbarg and N. S. Trüdinger, Elliptic partial differential equations of second order, Springer-Verlag, New York, 1977.
[Mi] J. Milnor, Morse theory, Ann. of Math. Stud., No. 51, Princeton University Press, 1963.
[Mo] C. Morry, Multiple integrals in the calculus of variations, Springer-Verlag, Heidelberg, 1966.
[O] Y. Otsu, On manifolds of positive Ricci curvature with large diameter, Math. Z., 206 (1991), 255-264.
[P] S. Peters, Convergence of Riemannian manifolds, Compositio Math., 62 (1987), 3-16.
[S] T. Sakai, On the continuity of injectivity radius function, Math. J. Okayama Univ., 25 (1983), 91-97.
[Sh] K. Shiohama, A sphere theorem for manifolds of positive Ricci curvature, Trans. Amer. Math. Soc., 275 (1983), 811-819.
[SS] J. Schouten and D. Struik, On some properties of general manifolds relating to Einstein's theory of gravitation, Amer. J. Math., 43 (1921), 213-216.
[We] A. Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geom., 9 (1974), 513-517.
[Wu1] J. -Y. Wu, The volume/diameter ratio for positively curved manifolds, Michigan Math. J., 37 (1990), 235-239.
[Wu2] J. -Y. Wu, Convergence of Riemannian 3-manifolds under Ricci curvature bound, Amer. J. Math., 116 (1994), 1019-1029.
[Yg] C. T. Yang, Odd-dimensional Wiedersehen manifolds are spheres, J. Differential Geom., 15 (1980), 91-96.

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