Hypersurfaces whose tangent geodesics do not cover the ambient space

Mendonça, Sérgio; Mirandola, Heudson

doi:10.1090/S0002-9939-07-09282-9

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 $x(\Sigma )$.

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