Fundamentalform of gauss map in the general case 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
 A: Note that $S_{n+1}$ is to be defined by $\langle x,x\rangle = -1$ (instead of $+1$, as in the OP).  The answers are computed by means of the structure equations and are as follows:
Let the first and second fundamental forms of $f$ be given by $I_f = g_{ij}\ dx^idx^j$ and $I\!I_f = h_{ij}\ dx^idx^j$ in some local coordinates. For the Gauss map $G:M\to \text{Gr}_n^+(2,n{+}2)$, one then has
$$
I_G = \left(g_{ij} + g^{kl}h_{ik}h_{jl}\right)\ dx^idx^j.
$$
To interpret the second fundamental form, one first remembers that $\text{Gr}_n^+(2,n{+}2)$ has the structure of a Kähler manifold (it is the dual of the Hermitian symmetric space $\text{SO}(n{+}2)/\text{SO}(2)\text{SO}(n)$).  Then one observes that $G$ immerses $M$ into $\text{Gr}_n^+(2,n{+}2)$ as a Lagrangian submanifold (with respect to the Kähler form), so that the normal bundle of the immersion $G$ into $\text{Gr}_n^+(2,n{+}2)$ is naturally isomorphic to the tangent bundle of $M$.  This means that the second fundamental form, which is usually thought of as a quadratic form with values in the normal bundle, can be regarded as a cubic form.  Because the immersion is Lagrangian, this cubic form is symmetric.  Calculation with the structure equations then shows that one has
$$
I\!I_G = \nabla^{I_f}\left(I\!I_f\right).
$$
I.e., the second fundamental form of the Gauss map $G$ is the covariant derivative of the second fundamental form of $f$ with respect to the Levi-Civita connection of the first fundamental form of $f$.  (Note that the fact that this covariant derivative is indeed symmetric is simply the Codazzi equation for the immersion $f$.)
