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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
DOI: https://doi.org/10.1090/S0002-9939-07-09282-9
Published electronically: November 30, 2007
MathSciNet review: 2361882
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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)$.


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Additional Information

Sérgio Mendonça
Affiliation: Departamento de Análise, Instituto de Matemática, Universidade Federal Fluminense, Niterói, RJ, CEP 24020-140, Brasil
Email: mendonca@mat.uff.br, sergiomendoncario@yahoo.com.br

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
Email: heudson@impa.br

DOI: https://doi.org/10.1090/S0002-9939-07-09282-9
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.

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