Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem

Seaman, Walter

doi:10.1090/S0002-9947-1990-1005083-4

Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem

Author: Walter Seaman
Journal: Trans. Amer. Math. Soc. 321 (1990), 811-823
MSC: Primary 53C20; Secondary 53C55
DOI: https://doi.org/10.1090/S0002-9947-1990-1005083-4
MathSciNet review: 1005083
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Abstract: An existence and uniqueness theorem is proved for algebraic curvature tensors and then applied to yield a global geometric theorem for locally weakly quarter pinched Riemannian manifolds whose second Betti number is nonzero.

[1] M. Berger, Sur quelques variétés Riemanniennes suffisamment pincées, Bull. Soc. Math. France 88 (1960), 57-71. MR 0133781 (24:A3606)
[2] A. Besse, Einstein manifolds, Springer-Verlag, Berlin, Heidelberg, and New York, 1987. MR 867684 (88f:53087)
[3] J. Cheeger and D. Ebin, Comparison theorem in Riemannian geometry, North-Holland, Amsterdam, 1975. MR 0458335 (56:16538)
[4] D. Gromoll and K. Grove, A generalization of Berger's rigidity theorem for positively curved manifolds, Ann. Sci. Ecole Norm. Sup. 20 (1987), 227-239. MR 911756 (88k:53062)
[5] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201-211 MR 0500705 (58:18268)
[6] S. Helgason, Differential geometry, Lie groups and symmetric spaces, 2nd ed., Academic Press, New York, 1978. MR 514561 (80k:53081)
[H] D. Hulin, Pinching and Betti numbers, Ann. Global Anal. Geom. 3 (1985), 85-93. MR 812314 (87f:53043)
[7] H. Karcher, A short proof of Berger's curvature tensor estimates, Proc. Amer. Math. Soc. 26 (1970), 642-644. MR 0270304 (42:5194)
[8] S. Kobayashi, On compact Kähler manifolds with positive Ricci tensor, Ann. of Math. (2) 74 (1961), 570-574. MR 0133086 (24:A2922)
[9] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 2, Interscience, New York, 1969.
[10] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391-404. MR 0088770 (19:577a)
[11] W. Poor, Differential geometric structures, McGraw-Hill, New York, 1981. MR 647949 (83k:53002)