definition of the hyperbolic space In Jurgen Jost's book, there is an example about Hyperbolic space in Chapter 1.The author mentions that we can equip $\Bbb R^{n+1}$ with an inner product
$\langle x,y\rangle=x^1y^1+\cdots + x^ny^n-x^{n+1}y^{n+1}$,where $x=(x^1,\cdots, x^{n+1}),y=(y^1,\cdots, y^{n+1})$.
According to the definition of inner product,the following property should be satisfied:$\langle x,x\rangle\geq 0$,but the inner product defined above does not have positivity.
 A: When a bilinear form is symmetric and non-degenerate, but not necessarily positive-definite, we can say that we have just a scalar product, instead of an inner product. The thing is that Minkowski space $(\mathbb{R}^{n+1}, \langle\cdot,\cdot\rangle_L)$, where $$\langle x,y\rangle_L =x^1y^1+\cdots +x^ny^n-x^{n+1}y^{n+1},$$is not an Euclidean vector space, as $\langle\cdot,\cdot\rangle_L$ is not positive-definite. However, you can still have subspaces of $\mathbb{R}^{n+1}$ for which the restriction of $\langle\cdot,\cdot\rangle_L$ is positive-definite. This is the case for the tangent spaces of the hyperbolic space $\mathbb{H}^n = \{x\in \mathbb{R}^{n+1}\mid \langle x,x\rangle_L=-1\}$, given by $T_x\mathbb{H}^n = x^\perp$, where $\perp$ is taken with respect to $\langle\cdot,\cdot\rangle_L$. It is a general fact for spaces with Lorentzian scalar products that the orthogonal complement of a timelike vector is a spacelike subspace. The first section of my notes about this topic might be helpful.
