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)



Hypersurfaces whose tangent geodesics do not cover the ambient space

Authors: Sérgio Mendonça and Heudson Mirandola
Journal: Proc. Amer. Math. Soc. 136 (2008), 1065-1070
MSC (2000): Primary 53C42; Secondary 53C22
Published electronically: November 30, 2007
MathSciNet review: 2361882
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ x:\Sigma^n\rightarrow M^{n+1}$ be an immersion of an $ n$-dimensional connected manifold $ \Sigma$ in an $ (n+1)$-dimensional connected complete Riemannian manifold $ M$ without conjugate points. Assume that the union of geodesics tangent to $ x$ does not cover $ M$. Under these hypotheses we have two results. The first one states that $ M$ is simply connected provided that the universal covering of $ \Sigma$ is compact. The second result says that if $ x$ is a proper embedding and $ M$ is simply connected, then $ x(\Sigma)$ is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set $ A\subset M$ such that $ \bar A$ is a manifold with the boundary $ x(\Sigma)$.

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

  • [AF] Alencar, H., Frensel, K., Hypersurfaces whose tangent geodesics omit a nonempty set, Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, (1991), 1-13. MR 1173029 (93f:53048)
  • [Al] Alexander, S., Saddle points of compact hypersurfaces. Geometriae Dedicata, 6 (1977), no.3, 353-362. MR 0482547 (58:2613)
  • [Be] Beltagy, M., Immersions into manifolds without conjugate points, J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), no. 3, 265-271. MR 1104375 (92f:53057)
  • [HK] Hasanis, T., Koutroufiotis, D. A property of complete minimal surfaces, Trans. Amer. Math. Soc., 281 (1984), no. 2, 833-843. MR 722778 (85j:53064)
  • [Hp] Halpern, B., On the immersion of an $ n$-dimensional manifold in $ n+1$-dimensional Euclidean space, Proc. Amer. Math. Soc., 30 (1971), 181-184. MR 0286116 (44:3330)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C42, 53C22

Retrieve articles in all journals with MSC (2000): 53C42, 53C22

Additional Information

Sérgio Mendonça
Affiliation: Departamento de Análise, Instituto de Matemática, Universidade Federal Fluminense, Niterói, RJ, CEP 24020-140, Brasil

Heudson Mirandola
Affiliation: Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, CEP 22460-320, Brasil
Address at time of publication: Departamento de Engenharia e Ciências Exatas, Centro Universitário Norte do Espírito Santo, Universidade Federal do Espírito Santo, São Mateus, ES, CEP 29933-480, Brasil

Keywords: Riemannian manifold, hypersurface, geodesic, star-shaped set
Received by editor(s): November 3, 2006
Published electronically: November 30, 2007
Additional Notes: This work was partially supported by CNPq, Brasil
Dedicated: We dedicate this work to our beloved wives Cristina and Fabiola
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society