We identify the set $M_4(\Bbb R)$ of all real square matrices of order 4 with the Euclidean space $\Bbb R^{16}$. Let $M$ be the subspace of $M_4(\Bbb R)$ which consists of all matrices $A=(a_{ij})_{i,j=1,2,3,4} $that satisfies the following condictions:
- $rank(A)=1$
- $A$ is a symmetric martix
- $a_{ii}\ge0,i=1,2,3,4$
- $\sum_{1\le i,j\le4}(a_{ij})^2=1$
Let $x$ be the 4-dimension column vector,consider the following mapping: $ϕ: \Bbb R^4 \mapsto M_4(\Bbb R), x\mapsto xx^T$ Answer the following questions.
(1)Show that $\left.\phi\right\rvert_{\Bbb R^4-\{0\}}:\Bbb R^4-\{0\} \mapsto M_4(\Bbb R)$ is an immersion.
(2)Show that $M$ is a $C^{\infty}$ submanifold of $M^4(\Bbb R)$
(3) Find the ratio $vol(M)/vol(S^3)$. Here, $vol(M)$ and $vol(S^3)$ are the volume of $M$ and $S^3$ in $M_4(\Bbb R)$ and $\Bbb R^4$ respectively.
For(1), I have calculated $J\phi=\begin{bmatrix} 2x_1& 0 & 0 & 0\\x_2 & x_1&0&0\\x_3&0&x_1&0\\x_4&0&0&x_1\\ \vdots&\vdots&\vdots&\vdots\end{bmatrix}$,suppose $x_1\neq0$,then $rank(J\phi)=4$,hence $\phi$ is an immersion.
For (2), define a equvalent relation on $\Bbb R^4 - \{0\} $ that $x$~$kx$ for some real number $k$, then $\Bbb R^4-\{0\}/$~$=\Bbb RP^3,$ then $\hat\phi:\Bbb RP^3\mapsto M$ is a continuous bijection, since $\Bbb RP^3$ is compact and $M$ is $T_2$ so it is a diffeomorphism. The projection is local diffeomorphism so by $(1)$, $\hat\phi$ is also an immersion, hence an embedding, so $M$ becomes a manifold diffeomorphic to $\Bbb RP^3$.
For $(3)$,$vol(M)/vol(S^3)=vol(\Bbb RP^3)/vol(S^3)=1/2$.
This is my proof but I don't know whether it's right or not, could you please tell me whether the proof is valid or not?