I have some question about a special type of hypersurfaces in manifolds. Let $M$ be a compact Riemannian manifold of nonpositive sectional curvature with convex boundary. We call two totally-geodesic immersions $\phi_1: M_1 \rightarrow M$ and $\phi_2: M_2 \rightarrow M$ of closed manifolds $M_1, M_2$ of nonpositive curvature in $M$ parallel, if there is a totally-geodesic embedding $\phi: N \times [a_1, a_2 ] \rightarrow \tilde{M}$ into the universal covering $\tilde{M}$ and covering maps $p_i: N \times \{a_i\} \rightarrow M_i$, which fulfill $\phi_i \circ p_i = \pi \circ \phi \vert_{N \times a_i} $ where $\pi: \tilde{M} \rightarrow M$ denotes the covering projection.

Now I have some questions: Let $F \subset \tilde{M}$ be a closed flat submanifold and denote by $\Gamma(F)$ the subgroup of the group $\Gamma$ of deck translations that fixes $F$. We call $F$ a $\Gamma$-flat if $\Gamma(F)$ acts cocompactly on $F$. Then the following seems to be true, but I don't know why and would be interested in a proof:

If $M$ is not flat, then the set of all flats parallel to a $\Gamma$-flat $F$ in $\tilde{M}$ is isometric to $F \times [a_1, a_2]$ for some compact intervall $[a_1, a_2]$.

If we assume that $[a_1, a_2 ]= [-a, a ]$ then we call the hypersurface $F \times 0$ in $F \times [-a, a ]$ central. Let $F'$ denote the immersed hypersurface in $M$ that is covered by $F$. Prove that every hypersurface parallel to $F'$ is homotopic to $F'$.

I would be very happy if someone could answer these questions!