Fundamentalform of gauss map in the general case

Question

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

Answer 1

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 $f$.)