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.
For clarity, I'll use hats and latic indices $\hat{g}_{ij}$, etc. to denote objects in the submanifold, and no hats and Greek indices $g_{\alpha\beta}$ to indicate objects in the ambient manifold. With the chosen coordinates the inclusion map is a graph: $$ x^0,x^1,\dots,x^{n+1}=f(\hat{x},r),\hat{x}^1,\dots,\hat{x}^{n+1} $$ The objects needed are the coordinate vectors $\hat{\partial}_i$ included into the ambient space (with some extension). These can be computed by chain rule. $$ \hat{\partial}_i=\frac{\partial x^\alpha}{\partial\hat{x}^i}\partial_\alpha=\frac{\partial f}{\partial x^i}\partial_0+\partial_i $$ Given $f(x,r)=\sqrt{\sum_{j=1}^{n+1}(x^j)^2-r^2}$, we can write this out more explicitly $$ \hat{\partial}_i=x^i\left(\sum_{j=1}^{n+1}(x^j)^2-r^2\right)^{-1/2}\partial_0+\partial_i $$ Technically, this is not a proper vector field and is is only defined on the submanifold. However, allowing $r$ to vary as a function of coordinates will provide a suitable extension, as it did for $\nu$.
Once you have these vector fields, as well as $\nu$, all of the computations in the formula $K_{ij}=g\left(\nabla_{\hat{\partial}_i}\nu,\hat{\partial}_j\right)$ take place in the ambient space.