Injectivity of map of fundamental groups from totally geodesic hypersurface Let $X$ be a compact manifold of non-positive sectional curvature which carries a connected totally geodesic hypersurface $X_0\subset X$. Let $K$ be any compact subset of $X-X_0$. That's to say we have $K\cap X_0=\emptyset$.
How can I prove that the induced homomorphism between fundamental groups $\pi_1 X_0\to \pi_1(X-K)$ is injective?
I have no idea about it. Could you please give me some help with the detail? thank you very much!
 A: You need $X_0$ to be closed inside $X$, otherwise, the theorem is false (there are simple counterexamples). So let us suppose that $X_0$ is closed.
Inclusions give the induced homomorphisms
$$\pi_1(X_0) \longrightarrow \pi_1(X\setminus K) \longrightarrow \pi_1(X)$$
In order to prove that the first homomorphism is injective, it suffices to prove that the composition $\pi_1(X_0) \to \pi_1(X)$ is injective. So you can forget about $K$.
To prove that $\pi_1(X_0)\to \pi_1(X)$ is injective, consider a point $p\in X_0$ and the exponential map
$$\exp_p\colon T_p X \longrightarrow X.$$
Since $X$ has non-positive curvature, this map is a covering (this is the Cartan-Hadamard Theorem). The hypersurface $X_0$ is closed, hence complete, hence geodesically complete by Hopf-Rinow Theorem. Therefore $X_0$ is the union of all the geodesics in $X_0$ starting from the point $p$. Since $X_0$ is a totally geodesic submanifold of $X$, the geodesics in $X_0$ are simply the geodesics in $X$, so finally we conclude that
$$X_0 = \exp_p(W)$$
where $W = T_pX_0$ is a vector hyperplane in $T_pX$. Since $W$ is simply connected and $\exp_p$ is the universal covering of $X$, one deduces that $X_0$ is $\pi_1$-injective in $X$.
