Manifolds of almost half of the maximal volume

Durumeric, Oguz C.

doi:10.1090/S0002-9939-1988-0958083-0

Manifolds of almost half of the maximal volume
HTML articles powered by AMS MathViewer

by Oguz C. Durumeric PDF

Proc. Amer. Math. Soc. 104 (1988), 277-283 Request permission

Addendum: Proc. Amer. Math. Soc. 107 (1989), 1145-1146.

Abstract:

The Riemannian manifolds with sectional curvature $\geq 1$ and volume slightly less (in terms of the dimension and the upper bound of the sectional curvature) than $V$, the half of the volume of the standard sphere, are classified. The behavior of the critical points of the distance function on manifolds whose volume differ from $V$ in terms of constructible constants in terms of the dimension is discussed.

References

Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
O. Durumeric, A generalization of Berger’s theorem on almost $\frac 14$-pinched manifolds. II, J. Differential Geom. 26 (1987), no. 1, 101–139. MR 892033

Rigidity of positively curved manifolds with large diameter

Detlef Gromoll and Karsten Grove, One-dimensional metric foliations in constant curvature spaces, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 165–168. MR 780042
Detlef Gromoll and Karsten Grove, A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 2, 227–239. MR 911756
Detlef Gromoll and Karsten Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988), no. 1, 143–156. MR 950559
Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
Karsten Grove and Peter Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), no. 1, 195–206. MR 951512, DOI 10.2307/1971439
Karsten Grove and Peter Petersen V, Homotopy types of positively curved manifolds with large volume, Amer. J. Math. 110 (1988), no. 6, 1183–1188. MR 970124, DOI 10.2307/2374690
Karsten Grove and Katsuhiro Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), no. 2, 201–211. MR 500705, DOI 10.2307/1971164
Stefan Peters, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), no. 1, 3–16. MR 892147