Hello, I have a question... I think it is worthy for MO.

Let $f: M \to S_{n+1}$ a spacelike immersion in the de Sitter-space with $S_{n+1}:=\lbrace X \in \mathbb{R}_{2}^{n+2}: \left\langle X, X \right\rangle=1\rbrace$.

Here is

$\mathbb{R}_{2}^{n+2}=(\mathbb{R}^{n+2}, \left\langle \cdot, \cdot \right\rangle)$

and

$\left\langle X, Y \right\rangle:=X_{1}Y_{1}+\ldots+X_{n}Y_{n}-X_{n+1}Y_{n+1}-X_{n+2}Y_{n+2}.$

We now consider the Gauss-map $G: M \to Gr_{n}^{+}(2, n+2)$ into Grassmannian manifold of all oriented spacelike n-planes in $\mathbb{R}_{2}^{n+2}$, where

$p \mapsto T_{p}M \in Gr_{n}^{+}(2, n+2)$.

Now denote with $g_{ij}$ the first and with $b_{ij}$ the second fundamentalform of $f$. Now the question ist:

**What is the first and second fundamentalform of $G$?**

I mean I have to calculate

$\tilde{g}_{\alpha\beta}\frac{\partial G^{\alpha}}{\partial x^{i}}\frac{\partial G^{\beta}}{\partial x^{j}}$.

But what is the metric $\tilde{g}$ in local coordinates of $Gr_{n}^{+}(2, n+2)$ and how can I calculate $\frac{\partial G^{\alpha}}{\partial x^{i}}$?

In the case $n=2$ it is easy, because $Gr_{2}^{+}(2, 4)$ is isometric to $\mathbb{H}^{2} \times \mathbb{H}^{2}$. But how does it work in the general case?

Many greetings Wolfgang