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Let be $SO(\mathbb{R^{n+2}})$ the special orthogonal group of $\mathbb{R^{n+2} }$, manifold of dimension $\frac{(n+1)(n+2)}{2}$, and $\mathbb{S^{n+1}}$ the canonical unit sphere of $\mathbb{R^{n+2}}$. Consider the following map

$$F:SO(\mathbb{R^{n+2}})\to \mathbb{S^{n+1}}$$

that, for each $Z\in SO(\mathbb{R^{n+2}}),$ $F(Z)$ is the last line of the matrix $Z$.

I would like to show that $F$ is a submersion and I thought in this way:

Thinking $SO(\mathbb{R^{n+2}})$ like a subset of $\mathbb{R^{(n+2)^2}}$, I can see the map $F$ as the projection of last $(n+2)$-coordinates, then $F$ is obviously linear so $dF_Z=F$, for all $Z\in SO(\mathbb{R^{n+2}})$.

Then we have that $$dF_Z=F:\mathbb{R^{(n+2)^2}}\to \mathbb{R^{n+2}}$$

so $dF_Z$ is surjective. Is that right?

Because, after that I would like to conclude, for a special $c\in \mathbb{S}^{n+1}$, that the manifold $F^{-1}(\{c\})$ has dimension $\frac{n(n+1)}{2}$, and I have some doubts about the derivative above.

I.e., if I can use this $$dF_Z=F:\mathbb{R^{(n+2)^2}}\to \mathbb{R^{n+2}}$$ with the argument above, or I should use

$$dF_Z:T_Z SO(\mathbb{R^{n+2}})\to T_{F(Z)}\mathbb{S^{n+1}}$$ with another argument to show that $F$ is a submersion.

Thank some help.


EDIT: By the Andrew D. Hwang comments, I need to show that the restriction of $dF_z$ into $T_Z SO^+(\mathbb{R}^{n+2})$ is surjective.

As $$T_Z SO(\mathbb{R}^{n+2})=\{X\in \mathcal{M}_{n+2}(\mathbb{R}) \ | \ X^t+X=0\}$$ and $$T_{F(Z)}\mathbb{S}^{n+1}=\{y\in \mathbb{R}^{n+2} \ | \ \langle F(z), y\rangle=0\},$$

given $x\in T_{F(Z)}\mathbb{S}^{n+1}$ I need to find $X\in T_Z SO(\mathbb{R}^{n+2})$ such that $F(X)=x$. Someone can help me?

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  • $\begingroup$ No, $F$ is not linear. In fact, your definition of $dF_Z$ doesn't seem correct. The tangent space of $\Bbb S^{n+1}$ is isomorphic to $\Bbb R^{n+1}$. Also, that of $SO(n+2)$, I think it should be $\Bbb R^{(n+2)^2-1}$. $\endgroup$ – Vim Jun 16 '17 at 21:34
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    $\begingroup$ @Irddo: You're right that $DF_{Z} = F$ is surjective as a map on $\mathbf{R}^{(n+2)^{2}}$, but to prove $F$ is a submersion you need to check that the restriction of $DF_{Z}$ to $T_{Z}SO(n+2)$ is surjective. $\endgroup$ – Andrew D. Hwang Jun 16 '17 at 21:59
  • $\begingroup$ @AndrewD.Hwang, thanks for your help. I will think how can I proove the condition to the restriction. Some help one more time? $\endgroup$ – Irddo Jun 17 '17 at 14:07

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