Assume that $M$ is a noncompact complete simply connected manifold of nonnegative sectional curvature. Then by Soul theorem, it has a soul $S$.

Question 1 :Fix a point $p\in S$. Then there is a totally geodesic submanifold of codimension 1 passing through $p$

example : In $\mathbb{E}^3$, $z=x^2+y^2$ has a non-selfintersecting infinite geodesic.

Question 2 :If $M$ is a noncompact complete Riemannian manifold and it contains aline$l$, then for any $p\in l$, $M$ has a totally geodesic submanifold $N$ of codimension 1 s.t. $N$ intersects the line transversally at $p$.