Hypersurfaces whose tangent geodesics do not cover the ambient space
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- by Sérgio Mendonça and Heudson Mirandola PDF
- Proc. Amer. Math. Soc. 136 (2008), 1065-1070 Request permission
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
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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
- Received by editor(s): November 3, 2006
- Published electronically: November 30, 2007
- Additional Notes: This work was partially supported by CNPq, Brasil
- Communicated by: Jon G. Wolfson
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2361882
Dedicated: We dedicate this work to our beloved wives Cristina and Fabiola