Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Remarks on sphere-type theorems


Authors: Hyeong In Choi, Sang Moon Kim and Sung Ho Park
Journal: Proc. Amer. Math. Soc. 125 (1997), 569-572
MSC (1991): Primary 53C20
MathSciNet review: 1343684
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove if $M$ is a complete Riemannian manifold with an embedded totally geodesic compact hypersurface $N$ such that $M$ has nonnegative sectional curvature, and the sectional curvature of $M$ is strictly positive in a neighborhood of $N$, then the pair $(M,N)$ is diffeomorphic to the pair $(S^n,S^{n-1})/\pi _1(M)$. This result gives an affirmative answer to a question of H. Wu in the case when $M$ is compact and simply connected.


References [Enhancements On Off] (What's this?)

  • 1. Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. MR 0458335
  • 2. T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68–73. MR 0187183
  • 3. R. E. Greene and H. Wu, 𝐶^{∞} approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376
  • 4. R. E. Greene and H. Wu, 𝐶^{∞} convex functions and manifolds of positive curvature, Acta Math. 137 (1976), no. 3-4, 209–245. MR 0458336
  • 5. H. Wu, On certain Kähler manifolds which are 𝑞-complete, Complex analysis of several variables (Madison, Wis., 1982) Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 253–276. MR 740887, 10.1090/pspum/041/740887
  • 6. H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525–548. MR 905609, 10.1512/iumj.1987.36.36029

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C20

Retrieve articles in all journals with MSC (1991): 53C20


Additional Information

Hyeong In Choi
Affiliation: Department of Mathematics, Seoul National University, Seoul, 151-742 Korea
Email: hichoi@math.snu.ac.kr

Sang Moon Kim
Affiliation: Department of Mathematics, Seoul National University, Seoul, 151-742 Korea

Sung Ho Park
Affiliation: Department of Mathematics, Seoul National University, Seoul, 151-742 Korea

DOI: https://doi.org/10.1090/S0002-9939-97-03480-1
Keywords: Sphere theorem, Morse theory, convex function
Received by editor(s): May 22, 1995
Additional Notes: Supported in part by BSRI-94-1416, and by GARC
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society