Nonnegatively curved contact manifolds

Goldberg, S. I.

doi:10.1090/S0002-9939-1986-0826497-3

Nonnegatively curved contact manifolds
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Proc. Amer. Math. Soc. 96 (1986), 651-656 Request permission

Abstract:

A compact simply connected contact Riemannian manifold with positive sectional curvature is homeomorphic with a sphere if its contact structure is normal and regular. In dimension 3 the regularity condition is omitted, and a stronger theorem is proved.

References

Kinetsu Abe and Joseph Erbacher, Nonregular contact structures on Brieskorn manifolds, Bull. Amer. Math. Soc. 81 (1975), 407–409. MR 417974, DOI 10.1090/S0002-9904-1975-13759-1
David E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, Vol. 509, Springer-Verlag, Berlin-New York, 1976. MR 0467588
W. M. Boothby and H. C. Wang, On contact manifolds, Ann. of Math. (2) 68 (1958), 721–734. MR 112160, DOI 10.2307/1970165
D. B. A. Epstein, Periodic flows on three-manifolds, Ann. of Math. (2) 95 (1972), 66–82. MR 288785, DOI 10.2307/1970854
S. I. Goldberg, Rigidity of positively curved contact manifolds, J. London Math. Soc. 42 (1967), 257–263. MR 232325, DOI 10.1112/jlms/s1-42.1.257
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
J. Martinet, Formes de contact sur les variétés de dimension $3$, Proceedings of Liverpool Singularities Symposium, II (1969/1970), Lecture Notes in Math., Vol. 209, Springer, Berlin, 1971, pp. 142–163 (French). MR 0350771
Frank Raymond, Classification of the actions of the circle on $3$-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78. MR 219086, DOI 10.1090/S0002-9947-1968-0219086-9

Almost contact manifolds

Yum Tong Siu and Shing Tung Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980), no. 2, 189–204. MR 577360, DOI 10.1007/BF01390043
Stephen Smale, The generalized Poincaré conjecture in higher dimensions, Bull. Amer. Math. Soc. 66 (1960), 373–375. MR 124912, DOI 10.1090/S0002-9904-1960-10458-2
Shûkichi Tanno, The topology of contact Riemannian manifolds, Illinois J. Math. 12 (1968), 700–717. MR 234486