Let G be the Minkovski quadratic form on $\mathbf{R}^{n+1}$ : $G(x,x)=-x_{0}^{2}+x_{1}^{2}+ \cdots +x_{n}^{2}$. Consider the (half) hyperboloid in $\mathbf{R}^{n+1}$ : $H=\{ x \in \mathbf{R}^{n+1} : G(x,x)=-1 \quad and \quad x_{0} >0\}$. For $p \in H$, the tangent space $T_{p}H$ can be considered a hyperplane of $\mathbf{R}^{n+1}$ and we define $g: T_{p}H \times T_{p}H \rightarrow \mathbf{R}$ by $g (x,y)=-x_{0}y_{0}+x_{1}y_{1}+\cdots +x_{n}y_{n}$. How to show that $(H,g)$ is a Riemannian manifold?
The only problem here is to show the positivity of $g$, $\quad i.e. \forall v \in T_{p}H$, $v \not= 0$, $g(v,v)>0.$
Could anybody help on this? Thank you!