# 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

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nobody an idea? – Wolfgangm Dec 5 '11 at 15:24
@Wolfganm: Well, some of us do know the answer and also know that it can be found in most good differential geometry books. I'm a little concerned that you have not carefully written out the second fundamental form $II_f$ of the map $f$. You've just written $b_{ij}$, but, of course, the second fundamental form is vector valued since you are in codimension $2$. If you aren't aware of that, then you probably should be looking for a good introductory book on submanifold theory. If you are already aware of that and just want a hint as to what the formula looks like, I can give you one. – Robert Bryant Dec 5 '11 at 18:16
Thanks Robert. I think I want a hint as to what the formula looks like. Can one written the second fundamentalform in dependence of the first and second fundamentalform $g_{ij}$ and $h_{ij}$ of $F$? – Wolfgangm Dec 7 '11 at 8:05
Or can you give me a book, where I can read something about? – Wolfgangm Dec 8 '11 at 14:41
@Wolfgangm: There are several books that treat pseudo-Riemannian geometry and submanifold theory. I don't know the best one to recommend for you because I don't know what you know. I can give you an answer to the above question, except that I now realize that there is something wrong with your question: It is not possible to have an $n$-dimensional spacelike submanifold of $S_{n+1}$ because the induced metric on this submanifold has type $(n{-}1,2)$. Did you mean to have $S_{n+1}$ be defined by $\langle x,x\rangle=-1$ (instead of $+1$)? With the minus, it would have type $(n,1)$. – Robert Bryant Dec 10 '11 at 14:33

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 $f$.)
Oh yes, sry. I mean $S_{n+1}:=\lbrace X \in \mathbb{R}_{2}^{n+2}: \left\langle X, X \right\rangle=-1\rbrace$. Sry. Greetings! – Wolfgangm Dec 11 '11 at 20:02
Ah, very cool! But, if you calculate the first fundamentalform you need $\mathcal{G}_{i}^{\alpha}$. Right? And for the second fundamental form too!? Thanks! – Wolfgangm Dec 12 '11 at 14:55
@Wolfgangm: Actually, if you do the computation via the moving frame (as I did), you don't need to know those particular quantities; you get the answers without having to write out the natural metric on $\text{Gr}^+(n,n{+}2)$. Also, I should point out that I'm regarding $\text{Gr}^+(n,n{+}2)$ as an abstract manifold, not as embedded into some vector space (similar to the way you are thinking of $S_{n+1}$ as an abstract manifold when you write the second fundamental form of $f$ as a quadratic form, rather than as a quadratic form with values in the normal bundle in $\mathbb{R}^{n+2}_2$. – Robert Bryant Dec 12 '11 at 15:57
Thanks for your efforts. I have two much questions: 1.) Can one seen that $\mathcal{G}$ is Lagrangian immersion in local coordinates? This follows from the Kahler property, too? 2.) It is very interesting that I do not need the metric or $\mathcal{G}_{i}^{\alpha}$ to calculate the fundamentalforms. But how do you mean "via the moving frame"? Sry for this stupid question. But I do not know. Greetings! – Wolfgangm Dec 13 '11 at 9:40