Given the Minkowski-metric $I$ with $I(x,x)=-x_0^2+x_1^2+...+x_n^2$. The set of upper plane of a two sheet hyperboloid is given by $\Bbb H:=\{x\in\Bbb R^{n+1}|I(x,x)=-1, x_0>0\}$. The corresponding tagent space at p is thus $T_p\Bbb H:=\{y\in \Bbb R^{n+1}|I(p,y)=0\}$.
Question: If I want to compute the tangent plane at $p=(1,0,...,0)$, acoording to the definition, I would obtain a plane at $x_0=0$. However, we would expect a plane at $x_0=1$ as the point $p$ has to lie in the tangent space $T_p\Bbb H$. Did I misunderstood something?