Show that any geodesics on the submanifold $H^n$ of Minkowski space $\R_1^{n+1}$ is the intersection of a two dimensional subspace of $\R_1^{n+1}$ with $H^n$.
Suppose $c(s)=(c_0(s),\ldots,c_n(s))$ is a geodesic in $H^n$, then
$$
X(s)\colon=c^\prime(s)=\sum c_j^\prime(s) E^j,\quad
X^\prime(s)=\sum c_j^\dprime(s) E^j,\quad
N(s)=\sum c_j(s)E^j
$$
and as $N(s)$ is time-like the geodesic equation gives
$$
0
=\bnabla X(s)
=X^\prime(s)+\la X^\prime(s),N(s)\ra N(s)
=X^\prime(s)+\Big(\sum\e_jc_j^\dprime(s)c_j(s)\Big)N(s)
$$
This is equivalent to the system of ODEs:
$$
\forall k=0,\ldots,n:\quad
c_k^\dprime+c_k\Big(\sum\e_jc_jc_j^\dprime\Big)=0~.
$$
Now we proceede as in exam: By exam we may assume that $\sum\e_jc_j^{\prime2}=1$. By definition of $H^n$: $\sum\e_jc_j^{2}=-1$, which implies: $\sum\e_jc_jc_j^{\prime}=0$ and therefore
$$
\sum\e_jc_jc_j^{\dprime}=-\sum\e_jc_j^{\prime2}=-1~.
$$
Hence $c$ is a unit speed geodesic if and only if
$$
\forall k=0,\ldots,n:\quad
c_k^\dprime-c_k=0
$$
Suppose again that $\sum\e_jc_j(0)n_j=\sum\e_jc_j^\prime(0)n_j=0$, then $f(s)\colon=\sum\e_jc_j(s)n_j$ solves the initial value problem $f^\dprime-f=0$, $f(0)=f^\prime(0)=0$, i.e. by uniqueness of solutions to ODE: $f=0$ and thus the curve $c$ lies in the two dimensional plane spanned by $(c_1(0),\ldots,c_n(0))$ and $(c_1^\prime(0),\ldots,c_n^\prime(0))$.