I am trying to practice computing the second fundamental form and the mean curvature, and I am trying to compute them for the Pseudo-Sphere in $n+1$ Minkowski spacetime.
Pseudo-Sphere in $n+1$ Minkowski spacetime is defined as: $\mathbb{S}^{1, n}(r):=\left\{x \in \mathbb{R}^{1, n+1} | \eta(x, x)=r^{2}\right\}$. As my first step I found the unit normal as $\nu = \frac{1}{r}x^{i}\partial_i$.
So I try using the definition of the second fundamental form $K_{ij}=g\left(\nabla_{\partial_{i}} \nu, \partial_{j}\right)$, and for the mean curvature its the trace of the second fundamental form.
But I am not sure how to follow through with the calculations. Any help is very appreciated.