We denote by arcch $:[1,+\infty)\rightarrow\mathbb{R}_+$ the inverse of the hyperbolic cosine. Let $v\in\mathcal{H}$. Show that the set $T_v\mathcal{H}$ of vectors tangent to $\mathcal{H}$ at point $v$ is a vector subspace of $V$ and determine this subspace. Deduce that the restriction of $B$ to $T_v\mathcal{H}$ is an inner product.