Hypersurfaces whose tangent geodesics do not cover the ambient space

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 $x(\Sigma)$.